**CHAN MATHEMATICS AND CHAN CODING AND CHAN CODE**

**H03M7/30**WO2009151893A2 | 2009-12-17 |

US20080018502A1 | 2008-01-24 | |||

US5384598A | 1995-01-24 | |||

US6094454A | 2000-07-25 |

Claims [1] A method of using CHAN MATHEMATICS in processing digital data or digital information using sums of and differences between digital values, basic or derived values, including the use in compressing digital information and decompressing the compressed code and restoring it to the original digital code of the original digital information input correctly and losslessly, including random data and data in even distribution; [2] The method of CLAIM [1] in making compression, including the following steps: (a) read in the original digital information, (b) analyze the digital information to obtain its characteristics, including the components of the Compression Unit and their relations, being expressed in sum of values between the components of the Processing Unit, (c) compute, through applying mathematical formula or formulae designed, which describe the characteristics of or the relations between the components of the original digital information obtained after analysis so that the characteristics of the original digital information are represented in the individual pieces of the resultant CHAN CODE, the number of digital bits of the compressed code, the CHAN CODE, being less than the number of digital bits used in the original code, whether in random or in even distribution or not; the compressed code being a lossless compression code that could be restored to the original code lossless on decompression; and (d) produce the corresponding compressed code resulting from the original digital information read in step (a); [3] The method of CLAIM [1] in making decompression, including the following steps for decompressing the compressed code and restoring it correctly and losslessly to the original digital code: (a) read in the corresponding compressed code, the CHAN CODE, including the RP Piece and the CV Piece at least, (b) obtain the characteristics of the corresponding compressed code, (c) apply in a reverse manner mathematical formula or formulae designed, which describe the characteristics of or the relations between the components of the original digital information obtained after analysis, to the compressed code using CHAN MATHEMATICS; (d) produce, after using step (c,) the original code of the original digital information from the corresponding compressed code correctly and losslessly, whether the original digital information is in random or in even distribution or not; [4] CHAN MATHEMATICS being a mathematical paradigm and the associated mathematical calculation logic and techniques used in merging and separating digital information; including values of Compression Units of a Processing Unit in compression and decompression of digital information, whether in random or in even distribution or not; [5] CHAN FORMULAE being formulae describing the characteristics and relations between basic components, the Compression Units and derived components such RP Piece of CHAN CODE and other derived components, such as the Combined Values or sums or differences of values of basics components of a Processing Unit, in making compression and decompression of digital information, whether in random or in even distribution or not; [6] CHAN CODING, using logic of CHAN MATHEMATICS and the associated CHAN FORMULAE, especially making compression and decompression of digital information, whether in random or in even distribution or not; [7] CHAN CODING techniques including any one of Rank Position Coding for making CHAN CODE, Absolute Address Branching for making CHAN CODE, Range Limiting, Base Shifting for making CHAN CODE, designing formulae for making CHAN CODE with reference to the characteristics and relationship amongst the basic components of the Processing Unit, and using such formulae for the processes of compression encoding and decompression decoding of digital information, whether in random or in even distribution or not; [8] CHAN SHAPES including CHAN DOT, CHAN LINES, CHAN TRIANGLE , CHAN RECTANGLES, CHAN TRAPEZIA AND CHAN SQUARES AND CHAN BARS representing the characteristics and relations of the basic components of a Processing Unit; [9] CHAN CODE being the resultant code of CHAN CODING, consisting at least any one of the RP Piece and the CV Piece; [ 10] CHAN CODE FILES, being digital information files containing CHAN CODE; [11] COMPLEMENTARY MATHEMATIC S using a constant value or a vari able containing a value as a COMPLEMENTARY CONSTANT or COMPLEMENTARY VARIABLE for mathematical processing, making the mirror value of a value or a range or ranges of values being obtainable for use; and [12] CHAN MATHEMATICS using COMPLEMENTARY MATHEMATICS and normal mathematics or either of them alone for processing. |

CHAN MATHEMATICS AND CHAN CODING AND CHAN CODE

Technical Field

[0] Let him that hath understanding count the number

[1] This invention claims priority of an earlier PCT Application, PCT/IB2016/054562 filed on 29 July 2016, submitted by the present inventor. This invention relates to the use of the concept and techniques revealed in the aforesaid PCT Application together with the relationship between different components of CHAN SHAPES (including CHAN RECTANGLES, CHAN TRAPESIA, CHAN SQUARES, CHAN TRIANGLE, CHAN LINE, CHAN DOT AND CHAN BARS or other shapes which describe the relations and characteristics of the basic components of the Processing Unit for use in making compression and decompression using CHAN MATHEMATICS for coding, CHAN MATHEMATICS being a combination of normal mathematics and

COMPLEMENTARY MATHEMATICS adopted for use in CHAN CODING) and the respective mathematical techniques to be revealed later in this Application, in making compression and decompression of digital information for the use and protection of intellectual property, expressed in the form of digital information, including digital data as well executable code for use in device(s), including computer system(s) or computer- controlled device(s) or operating-system-controlled device(s) or system(s) that is/are capable of running executable code or using digital data. Such device(s) is/are mentioned hereafter as Device(s).

[2] In particular, this invention relates to the method and schema as well as its application in processing, storing, distribution and use in Device(s) of digital information, including digital data as well as executable code, such as boot code, programs, applications, device drivers, or a collection of such executables constituting an operating system in the form of executable code embedded or stored into hardware, such as embedded or stored in all types of storage medium, including read-only or rewritable or volatile or non-volatile storage medium (referred hereafter as the Storage Medium) such as physical memory or internal DRAM (Dynamic Random Access Memory) or hard disk or solid state flash disk or ROM (Read Only Memory), or readonly or rewritable CD/DVD/HD-DVD/Blu-Ray DVD or hardware chip or chipset etc. The method of coding revealed, i.e. CHAN CODING, when implemented produces a compressed code, CHAN CODE that could be de-compressed and restored losslessly back into the original code; and such compressed code could also be re-compressed time and again until it reaches its limit.

[3] In essence, this invention reveals method and schema as well as its application that could be used to make compression of digital information. In this relation, it makes possible the processing, storing, distribution and use of digital information in Device(s) connected over local clouds or internet clouds for the purpose of using and protecting intellectual property. As with the use of other compression methods, without proper decompression using the corresponding methods, the compressed code could also considered an encrypted code as well. CHAN MATHEMATICS, CHAN CODING AND CHAN CODE (CHAN MATHEMATICS, CHAN CODING AND CHAN CODE including the concepts and techniques and the resultant code so produced as revealed in the aforesaid PCT Application, i.e. COMPLEMENTARY MATHEMATICS,

COMPLEMENTARY CODING AND COMPLEMENTARY CODE, and those to be revealed later in the present Application) could also be used in other scientific, industrial and commercial endeavors in various kinds of applications to be explored. The use of it in the Compression Field demonstrates vividly its tremendous use.

[4] However, the method and the schema as well as its application revealed in this

invention are not limited to delivery or exchange of digital information over clouds, i.e. local area network or internet, but could be used in other modes of delivery or exchange of information.

Background Art

[5] There are many methods and algorithms published for compressing digital information and introduction to commonly used data compression methods and algorithms could be found at http : //en . wikipedia. org/wiki/Data_compre ssi on . The present invention describes a novel method of making lossless data compression. Relevant part of the aforesaid wiki on lossless compression is reproduced here for easy reference:

"Lossless data compression algorithms usually exploit statistical redundancy to represent data more concisely without losing information, so that the process is reversible. Lossless compression is possible because most real-world data has statistical redundancy. For example, an image may have areas of colour that do not change over several pixels; instead of coding "red pixel, red pixel, ..." the data may be encoded as "279 red pixels". This is a basic example of run-length encoding; there are many schemes to reduce file size by eliminating redundancy.

The Lempel-Ziv (LZ) compression methods are among the most popular algorithms for lossless storage. [6] DEFLATE is a variation on LZ optimized for decompression speed and compression ratio, but compression can be slow. DEFLATE is used in PKZIP, Gzip and PNG. LZW (Lempel-Ziv-Welch) is used in GIF images. Also noteworthy is the LZR (Lempel-Ziv-Renau) algorithm, which serves as the basis for the Zip method. LZ methods use a table-based compression model where table entries are substituted for repeated strings of data. For most LZ methods, this table is generated dynamically from earlier data in the input. The table itself is often Huffman encoded (e.g. SHRI, LZX). A current LZ-based coding scheme that performs well is LZX, used in Microsoft's CAB format.

The best modern lossless compressors use probabilistic models, such as prediction by partial matching. The Burrows-Wheeler transform can also be viewed as an indirect form of statistical modelling. [7]

The class of grammar-based codes are gaining popularity because they can compress highly repetitive text, extremely effectively, for instance, biological data collection of same or related species, huge versioned document collection, internet archives, etc. The basic task of grammar-based codes is constructing a context-free grammar deriving a single string. Sequitur and Re-Pair are practical grammar compression algorithms for which public codes are available.

In a further refinement of these techniques, statistical predictions can be coupled to an algorithm called arithmetic coding. Arithmetic coding, invented by Jorma Rissanen, and turned into a practical method by Witten, Neal, and Cleary, achieves superior compression to the better-known Huffman algorithm and lends itself especially well to adaptive data compression tasks where the predictions are strongly context-dependent. Arithmetic coding is used in the bi-level image compression standard JBIG, and the document compression standard Dj Vu. The text entry system Dasher is an inverse arithmetic coder. [8]"

[6] In the aforesaid wiki, it says that "LZ methods use a table-based compression model where table entries are substituted for repeated strings of data". The use of table for translation, encryption, compression and expansion is common but how the use of table for such purposes are various and could be novel in one way or the other.

[7] The present invention presents a novel method, CHAN CODING, using CHAN

MATHEMATICS (i.e. using normal mathematical processing and

COMPLEMENTARY MATHEMATICS processing together) that produces amazing compression ratio that has never been revealed elsewhere. This represents a successful challenge and a revolutionary ending to the myth of Pigeonhole Principle in

Information Theory. CHAN CODING demonstrates how the technical problems described in the following section are being approached and solved.

Disclosure of Invention

Technical Problem

[8] The technical problem presented in the challenge of lossless data compression is how longer entries of digital data code could be represented in shorter entries of code and yet could be recoverable. While shorter entries could be used for substituting longer data entries, it seems inevitable that some other information, in digital form, has to be added in order to make it possible or tell how it is to recover the original longer entries from the shortened entries. If too much such digital information has to be added, it makes the compression efforts futile and sometimes, the result is expansion rather than

compression.

[9] The way of storing such additional information presents another challenge to the

compression process. If the additional information for one or more entries of the digital information is stored interspersed with the compressed data entries, how to differentiate the additional information from the original entries of the digital information is a problem and the separation of the compressed entries of the digital information during recovery presents another challenge, especially where the original entries of the digital information are to be compressed into different lengths and the additional information may also vary in length accordingly.

[10] This is especially problematic if the additional information and the compressed digital entries are to be recoverable after re-compression again and again. More often than not, compressed data could not be re-compressed and even if re-compression is attempted, not much gain could be obtained and very often the result is an expansion rather than compression.

[11] The digital information to be compressed also varies in nature; some are text files, others are graphic, music, audio or video files, etc. Text files usually have to be compressed losslessly, otherwise its content becomes lost or scrambled and unrecognizable.

[12] And some text files are ASCII based while others UNICODE based. Text files of

different languages also have different characteristics as expressed in the frequency and combination of the digital codes used for representation. This means a schema and method which has little adaptive or all embracing power could not work best for all such scenarios. Providing a more adaptive and flexible or an all embracing schema and method for data compression is therefore a challenge.

Technical Solution

[13] It has long been held in the data compression field that pure random binary numbers could not be shown to be definitely subject to compression until the present invention. To provide a schema and method for lossless compression that suits to random digital information of different types and of different language characteristics, one has to invent a way to compress random digital information and to recover it successfully.

[14] Using CHAN MATHEMATICS AND CHAN CODING, the random digital information to be compressed and recovered need not be known beforehand. The following diagram is used to explain COMPLEMENTARY MATHEMATICS AND COMPLEMENTARY CODING as revealed in the present invention and in the aforesaid PCT Application:

Diagram 1

COMPLEMENTARY MATHEMATICS AND LEGEND a and b are two pieces of digital information, each representing the reading of one Compression Unit, i.e. the content or the value of that Compression Unit, read one after another, for instance a is read as the first Compression Unit and b the second; a piece of digital information constitutes a Compression Unit, and two such Compression Units constitute a Processing Unit under the schema and method used in COMPLEMENTARY MATHEMATICS AND

COMPLEMENTARY CODING; for convenience and ease of computation, each Compression Unit is best of equal size for one cycle of compression process, using the same number scale without having to do scale conversion; the Compression Unit could be expressed and represented on any appropriate number scale of choice, including binary scale, octary, hexidecimal, etc.; the size of Compression Unit, Compression Unit Size, could be of any appropriate choice of size, for instance on binary scale, such as 4 bits or 8 bits or 16 bits or 32 bits or 64 bits or any bit size convenient for computation could be used as Compression Unit Size, i.e. a Compression Unit Size of 3 bits and upward will give a very impressive compression ratio, depending on the Compression Unit Size used, the bigger the Compression Unit Size used, the greater the compression ratio; using Compression Unit Size of 2 bits with the schema and method revealed in this invention appears to give a break even result, i.e. no gain or loss in the resultant code, and given the need of use of header for giving other useful information for correct decompression, the result would be an expansion of data equivalent to the size of the header used; the digital number or value of each Compression Unit represents the digital content of the Compression Unit, the digital number signifying the bit signs of all the bits of the Compression Unit; to achieve compression and make it recoverable using CHAN

CODING, more than 1 Compression Unit is to be used as one Processing Unit for making compression; the schema revealed in the aforesaid PCT Application, using two Compression Units making up one Processing Unit requires more data information to be added and this leads to an uncertainty in making definite compression for every Processing Unit, i.e. some expansion of data may result for certain Processing Units; so other choices could be considered as revealed in the present invention, using more complicated calculation, the choice here being four Compression Units making up one Processing Unit; other choices could also be considered, however as CHAN MATHEMATICS AND CHAN CODING makes recompression possible time and again easily, it is sufficient for the exploration to stop here for the time being; and to make the compression possible, one has to find out the relations between the Compression Units used; to show how COMPLEMENTARY

MATHEMATICS AND COMPLEMENTARY CODING works using two Compression Units as a demonstration of the concept and the techniques used, it is to be defined using mathematical formulae as follows: where a and b are the two Compression Units making up one

Processing Unit in COMPLEMENTARY CODING applied in the present schema, each being the digital number representing the content or values of the digital information conveyed in the Compression Unit; a being read before b; where a could be a bigger or lesser value than b, and one could use another two variable names to denote the ranking in value of these two Compression Units:

A, being the bigger value of the two Compression Units;

B, being the smaller value of the two Compression Units;;

and where a and b are equal in value, then the one read first read is to be A and the second one B; so A is bigger or equal in value than B; and so a could be A or B, depending its value in relation to b. where, in view of the above, a bit, the RP Bit (i.e. the Rank and Position Bit), has to be used to indicate whether the first Compression Unit read is bigger / equal or smaller in value then the second one; this bit of code therefore signifying the relation between the position and ranking of the values of the two Compression Units read; where to compress a and b, one could simply add the values of a and b together into one single value, using a bit size of Compression Unit Size plus One bit as follows:

Diagram 2

Before Compression as in Diagram 1, assuming Compression Unit Size of 64 bits, having two Compression Units of a Processing Unit:

Diagram 3

After Compression, the resultant compressed Code, the CHAN CODE, consisting of RP Piece and CV Piece:

where the RP Bit (1 bit), the first piece, the RP Piece of CHAN CODE and the combined value of a and b, A+B, (65 bits, i.e. 64 bits plus one, being bit size of Compression Unit Size plus One bit) , i.e. the second piece, the Combined Value Piece or Compressed Value Piece (the CV Piece), of CHAN CODE makes up the resultant Compressed CHAN CODE, excluding the header information necessary for indicating the number of compression cycles that has been carried out for the original digital information as well as necessary for remainder code processing. Such header information formation and processing has been mentioned in another PCT Patent Application, PCT/IB2015/056562, dated August 29, 2015 that has also been filed by the present inventor and therefore it is not repeated here.

People skilled in the Art could easily make use of header processing mentioned in the aforesaid PCT Patent Application or in other designs with the use of CHAN MATHEMATICS and CHAN CODING to produce the resultant CHAN CODE, i.e. the RP Piece and the CV Piece of the CHAN CODE with header information for decompression processing for the whole digital compressed digital data file creating using CHAN MATHEMATICS and CHAN CODING, the resultant compressed digital data file containing at least a RP Piece and a CV Piece of CHAN CODE, as to be revealed later in the present invention, the CV piece could be further sub-divided into sub- pieces when more Compression Units are to be used.

After finding out the relations of the components, the two Compression Units of the Processing Unit, i.e. the Rank and Position as well as the sum listed out in Paragraph [14] above, such relations are represented in the RP Piece and the CV Piece of CHAN CODE using the simplest mathematical formula, A+B in the CV Piece. The RP Piece simply contains 1 bit, either 0 or 1, indicating Bigger / Equal and Smaller in value of the first value a in relation to the second value b of the two Compression Units read in one Processing Unit.

[16] Using the previous example, and on the 64 bit personal computers prevalent in the

market today, if each Compression Unit of 64 bits on binary scale uses 64 bits to represent, there could be no compression possible. So more than 1 Compression Unit has to be used as the Processing Unit for each compression step made. A digital file of digital information has to be broken down into one or more Processing Units for making each of the compression steps made, and the compressed code of each of the Processing Units thus made constitutes each unit of CHAN CODE, consisting of one RP Piece and one CV Piece. The digital file of the digital information after compression using CHAN

CODING therefore consists of one or more than one unit of CHAN CODE, being the CHAN CODE FILE. The CHAN CODE FILE, besides including CHAN CODE, may also include, but not necessarily any remaining bits of original digital information which does not make up to 1 Processing Unit together with other added digital information representing the header or the footer which is usually used for identifying the digital information, including the check-sum and the signature as to when the decompression has to stop, or how many cycle of compression or re-compression made, or how many bits of the original uncompressed digital information present in the beginning or at the end or somewhere as indicated by the identifier in the header or footer. Such digital information left not compressed in the present compression cycle could be further compressed during the next cycle if required. This invention does not cover how such additional digital information is to be designed, to be placed and used. It just covers the CHAN CODE. CHAN CODE could also be divided into two or more parts to be stored, for instance, each of the two pieces of CHAN CODE may be separately stored into two separate digital data file in sequence for the use in decompression or for delivery for convenience or for security sake. The header or footer could also be stored in another separate digital data file and delivered for the same purposes. Files consisting such CHAN CODE and header and footer files used with CHAN CODE are all CHAN CODE FILES.

[17] CHAN CODE is the compressed code using CHAN CODING. CHAN CODING

produces compressed code out of the original code, using digital bits of digital information, i.e. the CHAN CODE, representing the compressed code which is less than the number of bits used in the original code, whether random or not or even in distribution or not. CHAN CODE represents the result of CHAN CODING, using the corresponding mathematical formulae, i.e. the value of the RP Piece and the addition operation for making calculation and compression, expressing the relations between the basic or derived components of the Processing Unit so that the original code could be restored to.

[18] CHAN CODE, as described above, obtained after the processing through using CHAN CODING, includes the digital bits of digital information, organized in one unit or broken down into sub-pieces, representing the content of the original digital information, whether random or not or even in distribution or not, that could be recovered correctly and losslessly. What CHAN CODING does for compression after the decision on the selection of the number scale used for computation, the bit size of the Compression Unit being made and the components for the Processing Unit (i.e. the number of the

Compression Units for one Processing Unit; the simplest case, being using just two Compression Units for one Processing Unit as revealed in the aforesaid PCT Application) and their relations being defined in mathematical formula or formulae and being implemented in executable code used in digital computer when employed, includes the following steps: (1) read in the original digital information, (2) analyze the digital information to obtain its characteristics, i.e. the components of the Compression Unit and their relations, (3) compute, through applying mathematical formula or formulae designed, which describe the characteristics of or the relations between the components of the original digital information so obtained after the analysis of CHAN CODING, that the characteristics of the original digital are represented in the CHAN CODE, the number of digital bits of which is less than the number of digital bits used in the original code, whether in random or in even distribution or not; the CHAN CODE being a lossless compression code that could be restored to the original code lossless on decompression; and (4) produce the corresponding CHAN CODE related to the original digital information read in step (1). What the CHAN CODING does for decompressing the corresponding CHAN CODE back into the digital original code includes the following steps: (5) read in the corresponding CHAN CODE, (6) obtain the characteristics of the corresponding CHAN CODE, (7) apply in a reverse manner mathematical formula or formulae so designed, which describe the characteristics of or the relations between the components of the original digital information so obtained after the analysis of CHAN CODING, to the CHAN CODE, using CHAN MATHEMATICS, including the use of normal mathematics and COMPLEMENTARY MATHEMATICS; (8) produce after using step (7) the original code of the original digital information lossless, whether the original digital information is random or not or even in distribution or not. So on decompression, CHAN MATHEMATICS is used and the compressed code, the CHAN CODE in Diagram 3 is restored correctly and losslessly to the original digital data code in Diagram 2.

To give an example of and explain how COMPLEMENTARY MATHEMATICS does, one could refer to the following Diagram:

Diagram 4

COMPLEMENTARY MATHEMATICS CC - A = A ^{C } and A ^{c } + A CC or

B ^{c }+ B CC and

(A ^{c } + B) (CC - A) + B where CC is Complementary Constant or Variable, being a Constant Value or

Variable Value chosen for the operation of COMPLEMENTARY MATHEMATICS, which is defined as using the Complementary Constant or Variable to make mathematical calculation or operation having addition and subtraction logics as explained in the present invention, depending on situation more than one Complementary Constant or Variable could be designed and use for different operations or purposes where necessary or appropriate;

A is the value being operated on, the example used here is the Rank Value A, A being bigger or equivalent in value to B in the present case of using two Compression Unit Values only; so in the first formula: CC - A = A ^{c } where CC minus A is equal to A Complement, i.e. denoted by A ^{c }, which is the Complementary Value of A, or a mirror value, using the respective Complementary Constant or Variable; for instance, let CC be a constant of the maximum value of the Compression Unit Size, such as 8 bits having 256 values; then CC is 256 in value; and let A be 100 in value, then A ^{c } is equivalent to 256 minus 100 = 156; and the reverse operation is therefore A ^{c } + A = CC, representing the operation of 100 + 156 = 256; and in the fourth formula, (A ^{c }+ B) = (CC - A) + B; and let B be 50, then A ^{c }+ B = (256 - 100) + 50 = 156 + 50 = 206.

Diagram 4 gives the logics of the basic operations of the COMPLEMENTARY

MATHEMATICS invented by the present inventor that is sufficient for making the decompression process to be introduced later. However, for more completion illustration of the addition and subtraction operations of COMPLEMENTARY MATHEMATICS, such logics are defined and explained in Diagram 5 below:

Diagram 5

More Logics of COMPLEMENTARY MATHEMATICS Defined:

CC - (A + B) = (A+B) ^{c } or = A ^{c } - B or = B ^{c } - A and

CC-(A-B) = A ^{C } + B and

CC - A + B is an ambivalent and confusing case; this should better be represented clearly as: either

(CC - A) + B = A ^{c } + B or

(CC-B) + A = B ^{C }+A so to further illustrate the above logics of the subtraction operations of

COMPLEMENTARY MATHEMATICS defined, let CC be 256, Abe 100 and B be 50, then

CC - (A+ B) = (A+B) ^{c } or = A ^{c } - B or = B ^{c } - A

i.e.256 -(100 + 50) = (100 + 50) ^{c } = 256 - 150= 106=A ^{C }-B= 156- 50= 106 =

= B ^{C }-A=206- 100= 106

and

CC-(A-B) = A ^{C } + B

i.e.256 - (100 - 50) = 256 - (50) = 206 = 156 + 50 = 206 and

(CC - A) + B = A ^{c } + B

i.e. (256 - 100) + 50 = 156 + 50 = 206 or

(CC-B) + A = B ^{C }+A

i.e. (256 - 50) + 100 = 206 + 100 = 306 Using the above logics of the addition and subtraction operations of

COMPLEMENTARY MATHEMATICS defined, one could therefore proceed with showing more details about how COMPLEMENTARY MATHEMATICS work in following Diagram 6:

Diagram 6

Operation on Data Values or Data Ranges using COMPLEMETARY MATHEMATICS Let CC be 256, Abe 100 and B be 50

(1) normal mathematical processing:

divide 150 by 2, i.e. get the average of A and B:

= (A+B)/2 = 1/2 A + 1/2B = 75; and since A is the bigger value in A+B; therefore = A - 1/2(A-B) = 100 - 1/2(100 - 50) = 100 - 1/2(50) = 100 - 25 = 75;

= B + 1/2(A-B) = 50 + 1/2(100 - 50) = 50 + 1/2(50) = 50 + 25 = 75;

(2) COMPLEMENTARY MATHEMATICS processing:

make an operation of (CC - A) + B, i.e. operating CC on A, not B:

= (CC - A) + B = A ^{C } + B = (256 - 100) + 50 = 156 + 50 = 206;

; noting that to do the operation in this step, A and B must be separated first; the step is meant to illustrate the operation of COMPLEMENTARY MATHEMATICS here

(3) CHAN CODING using CHAN MATHEMATICS (normal mathematical processing and COMPLEMENTARY MATHEMATICS processing): add the result of Step (1) to the result of Step (2), using A - 1/2(A-B):

= A ^{c } + B + A - 1/2(A-B) = A ^{c } + A + B - 1/2(A-B)

= CC + B - 1/2(A-B) = 256 + 50 - 1/2(100 - 50)

= 256 + 50 - 25

=256 + 25;

(4) CHAN CODING using CHAN MATHEMATICS :

subtract CC from the result of Step (3):

= [CC + B - 1/2(A-B)] - CC = B - 1/2(A-B)

= [256 + 50 - 25] - 256

= [50 - 25];

(5) CHAN CODING using CHAN MATHEMATICS :

add the result of Step (1) to Step (4), using B + 1/2(A-B):

= [B - 1/2(A-B)] + [B + 1/2(A-B)]

= 2B

= [50 - 25] + [50 + 25]

= 25 + 75

= 100 (6) normal mathematical processing:

divide 2B by 2 to get the value of B:

= 2B/2 = B

= 100/2 = 50

(7) normal mathematical processing:

get the value of A by subtracting B from A+B:

= A+B-B

= 150 - 50

= 100

The above serves to show the differences amongst normal mathematical processing, COMPLEMENTARY MATHEMATICS processing, and CHAN CODING using CHAN MATHEMATICS.

[22] COMPLEMENTARY MATHEMATICS performed in Step (2) above could only be made only after A and B are separated and known beforehand, therefore another piece of data information, i.e. (A-B) has to be added, so that A and B could be separated using the formulae (A+B) + (A - B) = 2* A and 2* A / 2 = A as well as (A+B) +(A - B) = 2*B and 2*B / 2 = B. And Step (2) just shows how COMPLEMENTARY MATHEMATICS works when operating on such basic components. Using the RP Bit, A and B after separation could be restored correctly to the position of first value and the second value read as a and b.

[23 ] COMPLEMENTARY MATHEMATIC S does not directly help to meet the challenge of the Pigeonhole Principle in Information Theory. However it does highlight the concept of the subtraction of a smaller range of data values from a bigger range of data values and the concept of a mirror value given a Complementary Constant or Value. If the smaller range is inside and placed in the middle of the bigger range of data values, then 2 smaller sub-ranges of data values are obtained in the logic of CC - (A - B) = A ^{c } + B. In this way, if one could not deduce by logic the value of B, at least B could be obtained in the combined value of (A ^{c } + B) and it is where to try to get the value of B or to eliminate the value of B to enable further calculation to go on in the hope of further finding out some useful values for meeting the challenge of Pigeonhole Principle in Information Theory. It is with this insight together with CHAN RECTANGLE AND TRAPEZIUM to be explained later in this invention that the challenge of Pigeonhole Principle in Information Theory is met with successful result. To confirm the end to the myth of the Pigeonhole Principle in Information Theory, the present invention uses four Compression Units making up one Processing Units as shown in the following Diagram 7:

Diagram 7

CHAN SHAPES

In most cases, the four basic components of a Processing Unit could be arranged into 3 Arms, i.e. the Long Arm, the Middle Arm and the Short Arms, with 2 pairs of basic components, representing the Upper Corner (being the pair of the two basic components with a bigger sum) and the Lower Corner (being the pair of the two basic components with a smaller sum) of the respective arms. However, in rare cases the values of these pairs happen to have same values in one way or anther, so that there may be less than 3 arms, such as only 2 arms or 1 arm or even becoming a dot shape. Therefore the distribution of the values of the four basic components of a Processing Unit could be represented in different CHAN SHAPES as follows:

CHAN DOT ·

This is where all four basic components have the same value;

CHAN LINES

There are 2 CHAN LINES as follows:

CHAN LINE 1 : The three arms all overlap together with the Short Arm having values [1+4] being the Upper Corner and [2+3] being the Lower Corner.

= [2+3

CHAN LINE 2: The three arms all overlap together with the Short Arm having values [2+3] being the Upper Corner and [1+4] being the Lower Corner.

CHAN TRIANGLE

There are 2 arms, the Long Arm and the Middle Arm, and the Short Arm becomes a dot as its pairs of values [1+4] and [2+3] are equal.

[ 1 +2] - [3+4]

CHAN RECTANGLES AND TRAPEZIA AND SQUARES

CHAN RECTANGLE 1 showing the incoming stream of data values of 4 Compression

CHAN RECTANGLE 2 showing the Ranking and Position of incoming stream values of 4 Compression Units The above CHAN RECTANGLE shows the first Compression Unit Value a, of the Processing Unit is B, the second in ranking amongst the four Compression Units; the second Compression Unit Value b is C, the third in ranking; the third Compression Unit Value c is A, the first in ranking; the fourth Compression Unit Value d is D, the last in ranking.

CHAN TRAPEZIA showing the relationship between the four basic components of the CHAN RECTANGLES

CHAN TRAPEZIUM 1

Upper Corners of the 3 arms are [1+2], [1+3] and [1+4] and

Lower Corners of the 3 arms are [3+4], [2+4] and [2+3].

[1 +2H3+4]

CHAN TRAPEZIUM 1 shows the relationship amongst the four basic components, the four values of the four Compression Units shown in CHAN RECTANGLE 2 where A is re-denoted by [1], B [2], C [3] and D [4]; and (A+B) = [1+2], (A - B) = [1 - 2], and the like in the same way. It could be seen that the four basic components of the Processing Unit [1] , [2] , [3] and [4] could be arranged into three arms, being [1+2] - [3+4] i.e. the Long Arm, [1+3] - [2+4] the Middle Arm and [1+4] - [2+3] the Short Arm. The sum of the values of [l]+[2]+[3]+[4] = [1+2+3+4] is always the same for all the three arms. The differences amongst the three arms upon re-distribution or re-arrangement of their value in pair combinations is reflected in the lengths, i.e. the differences in values between the upper corners and lower corners, of the three arms.

The Long Arm and Middle Arm always stay the same way. The Upper Corner and Lower Corner of the Short Arm however would swap depending on the value distribution of the four basic components. So there are two scenarios, either [1+4] is bigger in value the [2+3] as in CHAN TRAPEZIUM 1 or the other way round, which is represented in CHAN TRAPEZIUM 2 as follows:

CHAN TRAPEZR7M 2

Upper Corners of the 3 arms are [1+2], [1+3] and [1+4] and

Lower Corners of the 3 arms are [3+4], [2+4] and [2+3].

In CHAN TRAPEZnjM 1, the values of the Long Arm, the Middle Arm and the Short Arm could be redistributed as follows:

Long Arm = [1+2] - [3+4] = [1 - 4] + [2 - 3] = [1 - 3] + [2 - 4];

Middle Arm = [1+3] - [2+4] = [1 - 4] - [2 - 3] = [1 - 2] + [3 - 4]; and

Short Arm = [1+4] - [2+3] = [1 - 3] - [2 - 4] = [1 - 2] - [3 - 4]. In CHAN TRAPEZIUM 2, the values of the Long Arm, the Middle Arm and the Short Arm could also be redistributed as follows:

Long Arm = [1+2] - [3+4] [l-4] + [2-3] = [2-4] + [l-3];

Middle Arm [1+3] - [2+4] [1 - 4] - [2 - 3] = [3 - 4] + [1 - 2]; and

Short Arm = [2+3] - [1+4] [2-4]-[l-3] = [3-4]-[l-2].

So in CHAN TRAPEZIUM 1 and 2, the Long Arm is always equal to or bigger than the Middle Arm by 2*[2-3].

But because of the two possible scenarios of swapping in values of the Upper Corner and Lower Corner of the Short Arm, in CHAN TRAPEZRJM 1, the Long Arm is always equal to or bigger than the Short Arm by 2* [2 - 4] and the Middle Arm is always equal to or bigger than the Short Arm by 2* [3 - 4].

And in CHAN TRAPEZIUM 2, the Long Arm is always equal to or bigger than the Short Arm by 2*[1 - 3] and the Middle Arm is always equal to or bigger than the Short Arm by 2*[1 -2].

CHAN TRAPEZIUM 3 or CHAN SQUARE 1

This is where the Middle Arm overlaps with the Long Arm with Upper Corner and Lower Corner of the Short Arm being [1+4] and [2+3] respectively. If the two arms therein are not equal in length, it is a trapezium, otherwise it is a square:

[1+2] = [1+3]

[1+4]

[2+3]

[3+4] CHAN TRAPEZIUM 4 or CHAN SQUARE 2

This is where the Middle Arm overlaps with the Long Arm with Upper Corner and Lower Corner of the Short Arm being [2+3] and [1+4] respectively. If the two arms therein are not equal in length, it is a trapezium, otherwise it is a square:

[1 +2] = [1 +3]

[2+3]

[1 +41

[3+4] = [2+4]

CHAN TRAPEZIUM 5 or CHAN SQUARE 3

This is where the Short Arm overlaps with the Middle Arm with Upper Corner and Lower Corner of the Short Arm being [1+4] and [2+3] respectively. If the two arms therein are not equal in length, it is a trapezium, otherwise it is a square:

[1 +2]

[1 +3] = [1 +4]

[2+4] = [2+3]

5+4] CHAN TRAPEZIUM 6 or CHAN SQUARE 4

This is where the Short Arm overlaps with the Middle Arm with Upper Corner and Lower Corner of the Short Arm being [2+3] and [1+4] respectively. If the two arms therein are not equal in length, it is a trapezium, otherwise it is a square:

[ 1 +2]

[1 +3 ] =[2+3]

[2+ 4] = [1 +4]

[3+4]

To make data compression, the four values of the four basic components have to be represented by 1 CV Pieces consisting of 3 sub-pieces of values in addition to the RP Piece, which is used to indicate the relationship between the Position and Rank of the values of the 4 basic components as shown in the following Diagram 8:

Diagram 8

CHAN RECTANGLES showing details of the positions and ranking of the four incoming basic components and the compressed CHAN CODE

CHAN RECTANGLE 3 showing the Ranking and Position of incoming stream of data values of 4 Compression Units and the 64 bit size used CHAN RECTANGLE 4 CHAN CODE, the compressed code created by using CHAN

CODING showing details of the RP Piece and CV Piece

One very distinguishing characteristic of the present invention is the varying bit sizes of values of the 3 sub-pieces making up the CV Piece and RP Piece itself varying between 4 bit and 5 bit; and despite their varying bit sizes, CHAN CODING techniques to be revealed later could be used to decompress the relevant CHAN CODE and restore it losslessly and correctly back into the original incoming digital data codes. The varying bit sizes used are intended for further raising the compression ratio through using CHAN CODING techniques over the compression ratio that could be achieved using CHAN FORMULAE of CHAN MATHEMATICS to be revealed later.

The RP Piece is to be explained here first. RP Piece is used for indicating the relative positions of the 4 Ranked Values of the four basic components, the four Compression Units, of a Processing Unit because the Ranking of the four basic components may vary with their positions, there is no fixed rule for determining the relationship between position and ranking of the values of the four basic components. There are altogether 24 combinations between Position and Ranking as shown in the following Diagram 9:

Diagram 9

Rank Position Code Table

Possible Combinations of Positions and Ranks of the 4 Basic Components

1 3 4 2

1 4 2 3

1 4 3 2

2 1 3 4

2 1 4 3

2 3 1 4

2 3 4 1

2 4 1 3

2 4 3 1

3 1 2 4

3 1 4 2

3 2 1 4

3 2 4 1

3 4 1 2

3 4 2 1

4 1 2 3

4 1 3 2

4 2 1 3

4 2 3 1

4 3 1 2

4 3 2 1

As there are altogether 24 variations between Rank and Position of the values of the four basic components in combination, one normally would have to use 5 bits to house and indicate these 24 variations of Rank and Position Combination so that on decomposition, the correct Rank and Position of the values of the four basic components could be restored correctly, i.e. the four rank values of the basic components could be placed back into their correct positions corresponding to the positions of these values in the incoming digital data input. However, a technique called Absolute Address Branching could be used to avoid wasting in space for there are 32 seats for housing only 24 variations and 8 seats are left empty and wasted if Absolute Address Branching is not to be used.

[28] To use the simplest case, one could have only 3 values, then normally 2 bits have to be use to provide 4 seats for the 3 variations of values. However, with Absolute Address Branching is used, for the case where value = 1, only 1 bit is used and for the case where the value = 2 or = 3, 2 bits however have to be used. For instance, the retrieving process works as follows: (1) read 1 bit first; (2) if the value is 0, representing the value being 1, then there is no need to read the second bit; and if the value is 1, then the second bit has to be read, if the second bit is 0, it represents that the value is 2 and if the second bit is 1, then the value is 3. So this saves some space for housing the 3 values in question. 1/3 of the cases or variations uses 1 bit and the other 2/3 of the cases or variations has to use 2 bits for indication.

[29] So using Absolute Address Branching, 8 variations out of the 24 variations require only 4 bits to house and the remaining 16 variations require 5 bits. That means, 4 bits provide only 16 seats and 5 bits provide 32 seats. And if there are 24 variations, there are 8 variations over the seats provided by 4 bits, so 8 seats of the 16 seats provided by 4 bits have to reserved for representing 2 variations. So one could read 4 bits first, if it is found that the value is between 1 to 8, then one could stop and does not have to read in anther bit. However, if after reading 4 bits and the value is between 9 to 16, for these 8 variations, one has to read in another bit to determine if the which value it represents, for instance after 9 is determined, it could represent 9 or another value such as 17, then one has to read in another bit, say if it is 0, that means it stays as 9 and if it is 1, then it is of the value of 17, representing a Rank Position Code having a value of 17, indicating the RP pattern that the values of [1], [2], [3] and [4] have to be put into the positions of 3,4,1 and 2 correspondingly by referring to and looking up the Rank Position Code Table in Diagram 9 above. Absolute Address Branching is therefore a design in which an address, instead of indicating one value as it normally does, now could branch to identify 2 or more values using extra one bit or more bits, depending on design.

[30] It now comes to the determination of the ranked values of the four basic components, A=[l], B=[2], C=[3] and D=[4]. To determine the values of [1], [2], [3] and [4], one could use formulae with respect to the CHAN RECTANGLES AND CHAN TRAPEZIA to represent the essential relations and characteristics of the four basic components where the RP Piece as explained in Paragraph [29] above and the CV Piece altogether takes up a bit size less than the total bit size taken up by the 4 incoming basic components, a, b, c and d, i.e. 4 times the size of the Compression Unit for a Processing Unit under the schema presented in the present invention using CHAN RECTANGLES AND CHAN TRAPEZIA as presented above. [31] After meticulous study of the characteristics and relations between the four basic components making up a Processing Unit represented in CHAN RECTANGLES AND CHAN TRAPEZIA and countless trial and error, the following combinations of formulae represented in 3 sub-pieces of the CV Piece is found to be working and good for illustrating the principle at work behind. There could be other similar formulae to be found and use. So there is no limit to, but including using the formulae presented below with reference to CHAN RECTANGLES AND CHAN TRAPEZIA. So the answer is:

(1) = ([l] - [4])

(2) = ([2] - [3])

(3) = ([3] + [4])

The above 3 values represented in the formulae of Step (1) to Step (3) are put into the three sub-pieces of the CV Piece of CHAN CODE during the compression process. These three values are stored into the CHAN CODE FILE as the three sub-pieces of the CV Piece together with the corresponding RP Piece upon compression. During decompression, the RP Piece and the CV Piece are read out for decompression by using Absolute Address Branching technique and by looking up the retrieved value, the Rank Position Code of the corresponding Processing Unit against the Rank Position Code to determine where the ranked values of [1], [2], [3] and [4] of the Processing Units are to be placed during decomposition. The ranked values of [1], [2], [3] and [4] are determined as shown in the following steps using the values of the 3 sub-pieces of the corresponding CV Pieces stored in Step (1) to Step (3) above. How the 3 sub-pieces of the CV Piece are to be placed will be revealed later after discussing the steps for the determination of the values of [1], [2], [3] and [4].

[32] These 4 ranked values of the corresponding Processing Unit are determined as follows:

(4) = (l) + (2)

^{= } ([1] ~ [4]) + ([2] - [3]); upon re-arrangement or re-distribution of these 4 ranked values, leading to;

= ([!] + [2]) - ([3] + [4]); the Long Arm obtained;

^{= } ([1] ~ [3]) + ([2] - [4]); for comparing the difference in length with other arms;

(5) = (l) - (2)

= ([l] - [4]) - ([2] - [3]);

= ([1] + [3]) - ([2] + [4]); the Middle Arm obtained;

= ([l] - [2]) + ([3] - [4]);

(6) = (l) + (3)

= ([l] - [4]) + ([3] + [4]);

= ([!] + [3]); the Upper Corner of the Middle Arm (2) + (3)

([2] - [3]) + ([3] + [4]);

([2] + [4]); the Lower Corner of the Middle Arm

(6) + (7)

([1] + [3]) + ([2] + [4]); being the sum of [1]+ [2] + [3] + [4], very useful for finding the Upper Corner of the Long Arm;

([1] + [2] + [3] + [4]);

(9) = (8) - (3)

= ([1] + [2] + [3] + [4]) - ([3] + [4]); (3) = [3 + 4] given as the Lower Corner of the

Long Arm;

^{= } ([1] ^{+ } [2]); the Upper Corner of the Long Arm;

As the normal mathematical processing does not seem to yield another further information for providing a solution to the challenge, one could try using

COMPLEMENTARY MATHEMATICS for further processing. Before further processing is to continue, one could explore what COMPLEMENTARY MATHEMATICS could contribute to the present endeavor using the following Diagram 10:

Diagram 10

CHAN BARS

Additional Unknown Information provided by COMPLEMENTARY MATHEMATICS

CHAN BAR 1

CC [1]

CHAN BAR 2

[2] ^{c } Unknown [2] - [3] known Part of [3] + [4] known

CC [1] [2] [3] [4]

From the above Diagram, it could be seen that there are three pieces of data which are known from Steps (1) to (3), including([l] - [4]), ([2] - [3]) and ([3] + [4]). Without using COMPLEMENTARY MATHEMATICS, in this case using Compression Unit Size as the Complementary Constant (CC), [l] ^{c } and [2] ^{c } are unknown. While ([1] - [4]) + ([2] - [3]) is the Long Arm and ([1] - [4]) - ([2] - [3]) is the Middle Arm, Short Arm is missing as well. So to find out the unknown data, COMPLEMENTARY MATHEMATICS could be used as follows:

(10) Let CC be the Compression Unit Size, i.e.64 bits;

= 2*CC - ([1 - 4] + [2 - 3] + [3 + 4]); see Diagram 10;

= [l] ^{c } + [2] ^{c }; COMPLEMENTARY MATHEMATICS giving mirror values

= [l] ^{c } + [l] ^{c }+([l]-[2]);as[2] ^{c }=[l] ^{c }+[l-2]; seeDiagram lO;

(11) = (10) + (2)

= [l] ^{c } + [l] ^{c }+[l]-[2] + [2]-[3];

= [l] ^{c } + [i] ^{c }+ [1] - [3];

(12) =(11)-(1)

= [l] ^{c } + [l] ^{c }+([l]-[3])-([l]-[4]);

= [l] ^{c } + [l] ^{c }+[l]-[3]-[l] + [4];

= [l] ^{c } + [l] ^{c }-[3] + [4];

= [l] ^{c } + [l] ^{c }-([3]-[4]);

(13) = (12) -(3)

= [If + [l] ^{c }-([3]-[4])-([3] + [4]);

= [I]' + [l] ^{c }- ([3] - [4]) - (([4] + [4]) + ([3] - [4]));

= ([I]' + [l] ^{c } - [4] - [4]) - ([3] - [4]) - ([3] - [4]);

(14) = (13)/2

= (([I]' + [l] ^{c } - [4] - [4]) - ([3] - [4]) - ([3] - [4]))/2;

= ([l] ^{c }-[4])-([3]-[4]);

= ([!] + [4]) -([3] -[4]);

= [l]-[3] + [4] + [4];

(15) =(14) + (6)

= [1] - [3] + [4] + [4] + ([1] + [3]);

= [1] + [1] + [4] + [4];

(16) = (15)/2

= ([!] + [!] + [4] + [4])/2;

^{= } ([1] + [4]); one corner of the Short Arm; (17) = (1) + (16)

= ([l] - [4]) + ([l] + [4]);

= [i] + [i];

(18) = (17)/2

= ([i] + [i])/2;

= [l]; VOILA;

(19) = (16) - (18)

= ([i] + [4]) - [i];

= [4];

(20) = (7) - (19)

= ([2] + [4]) - [4];

= [2]; and

(21) = (20) - (2)

= [2] - ([2] - [3]);

= [3]·

[35] It could therefore be seen that using very simple logic of CHAN MATHEMATICS,

including normal mathematical processing and COMPLEMENTARY MATHEMATCS as well as the additional information made available and shortened processing steps made possible (as compared with using normal mathematical processing alone) by using COMPLEMENTARY MATHEMATICS, tremendous progress has been made in the field of Compression and Decompression Science and Art. The end to the myth of the

Pigeonhole Principle in Information Theory as announced in PCT/IB2016/054562 is hereby confirmed. However, it is envisaged that COMPLEMENTARY MATHEMATICS and CHAN MATHEMATICS could be put into fruitful use in other scientific and commercial pursuits and endeavors as well.

[36] After determining the ranked values of [1], [2], [3] and [4] and using the RP Piece, the original digital data input of the corresponding Processing Unit could now be

decompressed losslessly and restored correctly into the right positions. The placement and the bit size used for storing the code represented by the formulae for Step (1) to Step (3) as the 3 sub-pieces of the CV Piece of the CHAN CODE could now be considered and further optimized for compression and storage during the compression process. It uses the concept of range limit.

[37] To consider which sub-piece of the 3 sub-pieces of the CV Piece of the CHAN CODE to be put first, one could consider if placing one sub-piece could give information for the placement of the other sub-piece to come so that storage space could be reduced. So by comparing [1] - [4], [2] - [3] and [3] + [4], it could be seen that normally 64 bits have to be used for each of the values of [1] - [4] and [2] - [3] and 65 bits for [3] + [4] so that there would not be overflowing from the bit size storage assigned. It seems giving 64 bits to [1] - [4] and 65 bits to [3] + [4] is unavoidable. However, because [2] - [3] is a value smaller and inside the range of [1] - [4], therefore it is obvious that [1] - [4] could be used as the range limit or set as the maximum value that the value of [2] - [3] could attain. So if [1] - [4] is of the value of 1,000 instead of the maximum value of 2 ^{A }64, then 1000 could be used as the range limit for storing [2] - [3]. And also Absolute Address Branching could also be used so that the limit of 1,024 could be reduced exactly to 1,000 though in this case the saving is very small. The bit size used is either 10 bits or 9 bits instead of the 64 bits normally required. The placement of these 3 sub-pieces of CV Piece of the CHAN CODE could therefore remain as the Steps (1) to (3) revealed above. So it is apparent that the 3 sub-pieces of the CV Piece and so the CV Piece could also vary in size from one Processing Unit to another if the concept and techniques of Range Limit and Absolute Address Branching are also used for further optimization of the

compression to be made.

[38] Upon further reflection, it appears that COMPLEMENTARY MATHEMATICS provides a more convenient approach and more technical tools for solving the compression problem. However, its importance lies rather in the paradigm it provides, including range processing, mirror value, as well as base shifting, for instance, the base for indicating the value of [l] ^{c } becomes the value of CC, i.e. the maximum value of the Compression Unit Size chosen in the present example, instead of the base of Value 0 when doing normal mathematical processing.

[39] It does not however mean that COMPLEMENTARY MATHEMATICS is the only way to end the myth of Pigeonhole Principle in Information Theory. More sophisticated formulae may be designed to do away with the use of COMPLEMENTARY

MATHEMATCS. Using the characteristics and relations as revealed in CHAN SHAPES, one may design formulae that could meet the challenge of Pigeonhole Principle in Information Theory using CHAN CODING with just normal mathematical processing. So CHAN MATHEMATICS AND CHAN CODING is a super set of mathematical processing including normal mathematics and COMPLEMENTARY MATHEMATICS used in conjunction or alone separately with reference to CHAN SHAPES and the characteristics and relations so revealed in making compression encoding and

decompression decoding of digital information, whether in random or in even

distribution or not.

[40] All in all, the conclusion is again:

Let him that hath understanding count the number

Advantageous Effects

[41] As there is no assumption about the incoming digital information, any numbers,

including random numbers or numbers in even distribution or not, could be compressed and re-compressed. In the present days of the era of information explosion, a simple and fast method of compression and re-compression is a blessing to the whole mankind in employing computers to compress more and more digital information using less and less storage space in every aspect of life. It surely could contribute to the effort of man-space exploration.

Best Mode

[42] Limited by the computer and the computing power available, increasing the size of the Compression Unit appears to offer the more and more compression ratio and thus is the best; whereas adjusting the Compression Unit Size to a lower value from one cycle to another could also help to further compress the file.

Mode for Invention

[43] It is revealed that different sizes of Compressing Unit using CHAN CODING produces different compression ratios in one cycle of compression. The limit of CHAN CODING for compression is digital information less than the size of Processing Unit on the number scale adopted. Which mode to use is therefore a matter of choice, depending on the computing power available, other considerations such as economy, marketing

considerations, etc. However, as re-compression could easily be made, it is insignificant to make the distinction.

[44] In summary, the method and schema revealed in the present invention is characterized by:

(1) A method of using CHAN MATHEMATICS in processing digital data or digital information using sums of and differences between digital values, basic or derived values, including the use in compressing digital information and decompressing the compressed code and restoring it to the original digital code of the original digital information input correctly and losslessly, including random data and data in even distribution;

(2) The method of (1) in making compression, including the following steps: (a) read in the original digital information, (b) analyze the digital information to obtain its characteristics, including the components of the Compression Unit and their relations, being expressed in sum of values between the components of the Processing Unit, (c) compute, through applying mathematical formula or formulae designed, which describe the characteristics of or the relations between the components of the original digital information obtained after analysis so that the characteristics of the original digital information are represented in the individual pieces of the resultant CHAN CODE, the number of digital bits of the compressed code, the CHAN CODE, being less than the number of digital bits used in the original code, whether in random or in even distribution or not; the compressed code being a lossless compression code that could be restored to the original code lossless on decompression; and (d) produce the corresponding compressed code resulting from the original digital information read in step (a);

(3) The method of (1) in making decompression, including the following steps for decompressing the compressed code and restoring it correctly and losslessly to the original digital code: (a) read in the corresponding compressed code, the CHAN CODE, including the RP Piece and the CV Piece at least, (b) obtain the characteristics of the corresponding compressed code, (c) apply in a reverse manner mathematical formula or formulae designed, which describe the characteristics of or the relations between the components of the original digital information obtained after analysis, to the compressed code using CHAN MATHEMATICS; (d) produce, after using step (c,) the original code of the original digital information from the corresponding compressed code correctly and losslessly, whether the original digital information is in random or in even distribution or not;

(4) CHAN MATHEMATICS being a mathematical paradigm and the associated mathematical calculation logic and techniques used in merging and separating digital information; including values of Compression Units of a Processing Unit in compression and decompression of digital information, whether in random or in even distribution or not;

(5) CHAN FORMULAE being formulae describing the characteristics and relations between basic components, the Compression Units and derived components such RP Piece of CHAN CODE and other derived components, such as the Combined Values or sums or differences of values of basics components of a Processing Unit, in making compression and decompression of digital information, whether in random or in even distribution or not;

(6) CHAN CODING, using logic of CHAN MATHEMATICS and the associated CHAN FORMULAE, especially making compression and decompression of digital information, whether in random or in even distribution or not;

(7) CHAN CODING techniques including any one of Rank Position Coding for making CHAN CODE, Absolute Address Branching for making CHAN CODE, Range Limiting, Base Shifting for making CHAN CODE, designing formulae for making CHAN CODE with reference to the characteristics and relationship amongst the basic components of the Processing Unit, and using such formulae for the processes of compression encoding and decompression decoding of digital information, whether in random or in even distribution or not;

(8) CHAN SHAPES including CHAN DOT, CHAN LINES, CHAN TRIANGLE , CHAN RECTANGLES, CHAN TRAPEZIA AND CHAN SQUARES AND CHAN BARS representing the characteristics and relations of the basic components of a Processing Unit; (9) CHAN CODE being the resultant code of CHAN CODING, consisting at least any one of the RP Piece and the CV Piece;

(10) CHAN CODE FILES, being digital information files containing CHAN CODE;

(11) COMPLEMENTARY MATHEMATICS using a constant value or a variable containing a value as a COMPLEMENTARY CONSTANT or COMPLEMENTARY VARIABLE for mathematical processing, making the mirror value of a value or a range or ranges of values being obtainable for use; and

(12) CHAN MATHEMATICS using COMPLEMENTARY MATHEMATICS and normal mathematics or either of them alone for processing.

Industrial Applicability

[45] There are numerous industrial applications that could use CHAN CODING and its

related schema at an advantage, including all computer applications that process digital information, including all types of digital data, whether in random or in even distribution or not.

[46] The prior art for the implementation of this invention includes computer languages and compilers for making executable code and operating systems as well as the related knowledge for making application or programs; the hardware of any device(s), whether networked or standalone, including computer system(s) or computer- controlled device(s) or operating- system-controlled device(s) or system(s), capable of running executable code; and computer-executable or operating-system-executable instructions or programs that help perform the steps for the method of this invention. In combination with the use of the technical features contained in the prior art stated above, this invention makes possible the implementation of CHAN CODING using CHAN

MATHEMATICS for the compression and decompression of random digital information, including digital data and digital executable codes; and in this relation, is characterized by the following claims:

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