AMENDED CLAIMS received by the International Bureau on 11 January 2021 (11.01.2021)

1. (Replaces Claim 1) An encoder apparatus for use in a communication system for encoding of a source data block d into a transmitted code block x, the encoder apparatus comprising: a data inserter; and a transform encoder, wherein the encoder apparatus is configured according to a set of parameters (N, K, G_out, G_in, A, a), wherein N is a length of the transmitted code block x, K is a length of the source data block d, wherein G_out is an outer transform matrix and G_n is an inner transform matrix, wherein A is a data index set, wherein a is a fixed data block, wherein N and K are integers that satisfy 1 £ K < N, wherein the data index set A is a subset of (1,2, ··· , N} with a size \A\ = K, wherein the fixed data block a has a length N — K, wherein the outer transform matrix G_out is an N X N upper triangular matrix with an element (g_out)_i,j at an ith row and a j th column of G_out for each i = 1,2, ··· , N and j = 1,2, ··· , N, wherein (g_out)_i,j = 0 for each pair of integers (i,j) satisfying 1 £ j < i £ N, wherein (g_out)_i,i ¹ 0 for each i = 1,2, ···, N, wherein (g_out)_i,j ¹ 0 for at least one pair of integers (i,j) satisfying 1 £ i < j £ N, wherein the inner transform matrix G_n is an N X N lower triangular polar transform matrix given by a Kronecker power wherein the data inserter is configured to receive the source data block d and generate a data container block v by setting v_A = d and v_Ac = a, wherein v_A denotes the data part of v corresponding to the coordinates of v whose indices are in A, and wherein v_Ac denotes the part of v corresponding to coordinates of v whose indices are not in A, wherein the transform encoder is configured to receive the data container block v and generate a transmitted code block x by computing x = vG_out G_n, wherein the encoder apparatus is further configured to send the transmitted code block x to a channel in the communication system.

2. (Replaces Claim 2) The encoder apparatus of claim 1, wherein the transform encoder comprises: an outer transform encoder configured to receive the data container block v and generate an outer transform block u = vG_out, and 2 an inner transform encoder configured to receive the outer transform block u and compute the transmitted code block x = uG_n.

3. (Replaces Claim 3) The encoder apparatus of claim 2, wherein the outer transform matrix G_out is a Toeplitz matrix defined by a causal impulse response c = (c₀, c₁, ···, c_m), wherein m > 0, c_Q ¹ 0, and c_m ¹ 0.

4. (Replaces Claim 7) The encoder apparatus of claim 1, wherein the data index set A is chosen in accordance with a pointwise score function, wherein the pointwise score function is one of a Hamming score function, a mutual information score function, or a Gallager score function.

5. (Replaces Claim 8) A decoder apparatus for use in a communication system for receiving a received code block y from a channel and decoding the received code block y to generate a decoded source data block d as an estimate of a source data block d, wherein the received code block y comprises a noisy version of a transmitted code block x, the decoder apparatus comprising: an inner decoder; and an outer decoder, wherein the decoder apparatus is configured according a set of parameters (N, K, G_out, G_n, A, a), wherein N is a length of the transmitted code block x, K is a length of the source data block d, wherein G_out is an outer transform matrix and G_n is an inner transform matrix, wherein A is a data index set, wherein a is a fixed data block, wherein N and K are integers that satisfy 1 £ K < N, wherein the data index set A is a subset of {1,2, ··· , N} with a size \A\ = K, wherein the fixed data block a has a length N — K, wherein the outer transform matrix G_out is an N X N upper triangular matrix with an element (g_out)_i,j at an ith row and a j th column of G_out for each i = 1,2, ··· , N and j = 1,2, ··· , N wherein (g_out)_i,j = 0 for each pair of integers (i,j) satisfying 1 £ j < i £ N, wherein(g_out)_i,i ¹ 0 for each i = 1,2, ··· , N, and wherein (g_out)_i,j ¹ 0 for at least one pair of integers (i,j) satisfying 1 £ i < j £ N, wherein the inner transform matrix G_in is a N X N lower-triangular polar transform matrix given by a Kronecker power G_in = 3 wherein the transmitted code block x is related to the source data block d by a relation x = vG_outG_in, wherein v is a data container block such that v_A = d and v_A^c = a, wherein v_A denotes the part of v corresponding to the coordinates of v whose indices are in A, and wherein v_Ac denotes the part of v corresponding to the coordinates of v whose indices are not in A, wherein the inner decoder is configured to receive the received code block y, receive node metric requests from the outer decoder, calculate node metrics in accordance with the inner transform matrix G_n, and send calculated node metrics to the outer decoder, wherein the outer decoder is configured to send node metric requests to the inner decoder, receive calculated node metrics from the inner decoder, and calculate a decoded data container block v in accordance with the outer transform matrix G_out, the data index set A, and the fixed data block a, and wherein the decoder apparatus further configured to extract the decoded source data block d from the decoded data container block v by setting d = v_A wherein v_A denotes the part of v corresponding to the coordinates of v whose indices are in A, and send the decoded source data block d to a destination in the communication system.

6. (Replaces Claim 9) The decoder apparatus of claim 5, wherein the outer transform matrix G_out is a Toeplitz matrix defined by a causal impulse response c = (c₀, c₁, ..., c_N-1), wherein c₀ ¹ 0 and and c_m ¹ 0 for at least one integer m satisfying 1 £ m £ N — 1.

7. (Replaces Claim 10) The decoder apparatus of claim 5, wherein the data index set A is chosen in accordance with a pointwise score function, wherein the pointwise score function is one of a Hamming score function, a mutual information score function, or a Gallager score function.

8. (Replaces Claim 11) The decoder apparatus of claim 5, wherein the outer decoder calculates the decoded data container block by using a depth-first tree search algorithm.

9. (Replaces Claim 13) An encoding method for use in a communication system for 4 encoding of a source data block d into a transmitted code block x using an encoder apparatus including a data inserter and a transform encoder, the method comprising: configuring the encoder apparatus according to a set of parameters ( N , K, G_out, G_n, A, a), wherein N is a length of the transmitted code block x, K is a length of the source data block d, wherein G_out is an outer transform matrix and G_in is an inner transform matrix, wherein A is a data index set, wherein a is a fixed data block, wherein N and K are integers that satisfy 1 £ K < N, wherein the data index set A is a subset of (1,2, · · · , N} with a size IAI = K , wherein the fixed data block a has a length N — K, wherein the outer transform matrix G_out is an N x N upper triangular matrix with an element (g_out)_i,j at an ith row and a j th column of G_out for each i = 1,2, ··· , N and j = 1,2, ··· , N, wherein (g_out)_i,j = 0 for each pair of integers (i,j) satisfying 1 £ j < i £ N, wherein (g_out)_i,i ¹ 0 for each i = 1,2, ··· , N, wherein (g_out)_i,j ¹ 0 for at least one pair of integers (i,j) satisfying 1 £ i < j £ N, wherein the inner transform matrix G_n is an N X N lower triangular polar transform matrix given by a Kronecker power wherein n = log₂ N, the encoding method comprising: receiving, at the data inserter, the source data block d; generating, within the data inserter, a data container block v by setting V_A = d and v_Ac = a, wherein V_A denotes the part of v corresponding to the coordinates of v whose indices are in A, and wherein v_Ac denotes the part of v corresponding to the coordinates of v whose indices are not in Jl; receiving the data container block v at the transform encoder from the data inserter; generating, in the transform encoder, the transmitted code block x by computing x = and sending the transmitted code block x to a channel in the communication system.

10. (Replaces Claim 14) The encoding method of claim 9, wherein the transform encoder comprises: an outer transform encoder receiving the data container block v and generating an outer transform block u = vG_out; and an inner transform encoder receiving the outer transform block u and computing the transmitted code block x = uG_in. 5

11. (Replaces Claim 15) The encoding method of claim 10, wherein the outer transform matrix G_out is a Toeplitz matrix defined by a causal impulse response c = (c₀, c₁, ..., c_m) , herein m > 0, c_Q ¹ 0, and c_m ¹ 0.

12. (Replaces Claim 18) The encoding method of claim 9, wherein the data index set A is chosen in accordance with a pointwise score function, wherein the pointwise score function is one of a Hamming score function, a mutual information score function, or a Gallager score function.

13. (Replaces Claim 19) A decoding method for use in a communication system for receiving a received code block y from a channel and decoding the received code block y using a decode apparatus including an inner decoder and an outer decoder to generate a decoded source block d as an estimate of a source data block d, wherein the received code block y comprises a noisy version of a transmitted code block x, the decoding method comprising: configuring the decoder apparatus according to a set of parameters ( N , K, G_out, G_n, A, a), wherein N is a length of the transmitted code block x, K is a length of the source data block d, wherein G_out is an outer transform matrix and G_in is an inner transform matrix, wherein A is a data index set, wherein a is a fixed data block, wherein N and K are integers that satisfy 1 £ K < N, wherein the data index set A is a subset of (1,2, · · · , N} with a size \A\ = K , wherein the fixed data block a has a length N — K, wherein the outer transform matrix G_out is an N x N upper triangular matrix with an element (g_out)_i,j at an ith row and a ;th column of G_out for each i = 1,2, ··· , N and j = 1,2, ··· , N, wherein (g_out);_,; ⁼ 0 for each pair of integers (i,j) satisfying 1 £ j < i £ N, wherein (g_out)_i,i ¹ 0 for each i = 1,2, ··· , N, and wherein (g_out)_i,j ¹ 0 for at least one pair of integers (i, j) satisfying 1 £ i < j £ N, wherein the inner transform matrix G_in is a N X N lower-triangular polar transform matrix given by a Kronecker power wherein n = log₂ N, wherein the transmitted code block x is related to the source data block d by a relation x = vG_outG_in, wherein v is a data container block such that v_A = d and v_Ac = a, wherein v_A denotes the part of v corresponding to the coordinates of v whose indices are in A, and wherein v_Ac denotes the part of v corresponding to the coordinates of v whose indices are not in A; 6 receiving, at the inner decoder, the received code block y; sending node metric requests from the outer decoder to the inner decoder; receiving, at the inner decoder, the node metric requests from the outer decoder; calculating, in the inner decoder, node metrics in accordance with the inner transform matrix G_in; sending calculated node metrics from the inner decoder to the outer decoder; receiving, at the outer decoder, the calculated node metrics from the inner decoder; calculating, in the outer decoder, a decoded data container block v in accordance with the outer transform matrix , the data index set A, and the fixed data block a; extracting the decoded source data block d from a part v_A of the decoded data container block v, wherein v_A denotes the part of v corresponding to the coordinates of v whose indices are in A; and sending the decoded source data block d to a destination in the communication system.

14. (Replaces Claim 20) The decoding method of claim 13, wherein the outer transform matrix G_out is a Toeplitz matrix defined by a causal impulse response c = (c₀, c₁, ..., c_N-1) , wherein c₀ ¹ 0 and and c_m ¹ 0 for at least one integer m satisfying 1 £ m £ N — 1.

15. (Replaces Claim 21) The decoding method of claim 13, wherein the data index set A is chosen in accordance with a pointwise score function, wherein the pointwise score function is one of a Hamming score function, a mutual information score function, or a Gallager score function.

16. (Replaces Claim 22) The decoding method of claim 13, wherein the outer decoder calculates the decoded data container block v by using a depth-first tree search algorithm.