Methods and apparatus for systematic encoding of data in error correction coding using triangular factorization of generator matrix

1. A systematic encoder apparatus for use in a communication system for reliable transfer of a source data block (SDB), the systematic encoder apparatus configured for an outer transform matrix and an inner transform matrix, the systematic encoder apparatus comprising:

an inner encoder; and

an outer encoder,

wherein the inner encoder is configured to receive the SDB and generate an output constraint block (OCB) as an image of the SDB under an inverse of a submatrix of the inner transform matrix,

wherein the outer encoder is configured to receive a fixed data block (FDB) and the OCB and generate a transform output block (TOB) subject to a first systematic coding constraint that the TOB is an image of a transform input block (TIB) under the outer transform matrix, a second systematic coding constraint that the TOB contains the OCB transparently in a sub-block of the TOB, and a third systematic coding constraint that the TIB contains the FDB transparently in a sub-block of the TIB,

wherein the inner encoder is further configured to receive the TOB and generate a transmitted code block (TCB) subject to a fourth systematic coding constraint that the TCB is an image of the TOB under the inner transform matrix and a fifth systematic coding constraint that the TCB contains the SDB transparently in a sub-block of the TCB.

2. The systematic encoding apparatus of claim 1 , wherein the outer transform matrix is an invertible upper triangular (IUT) matrix,

wherein the sub-block of the TOB comprises elements of the TOB whose indices belong to an index set, and

wherein the sub-block of the TIB comprises elements of the TIB whose indices belong to a complement of the index set.

3. The systematic encoding apparatus of claim 2 , wherein the inner transform matrix is an invertible lower triangular (ILT) matrix,

wherein the submatrix of the inner transform matrix comprises elements of the inner transform matrix whose column indices belong to the index set and whose row indices belong to the index set, and

wherein the sub-block of the TCB comprises elements of the TCB whose indices belong to the index set.

4. The systematic encoder apparatus of claim 2 , wherein the outer encoder comprises a feed-forward computation circuit, and

wherein the feed-forward computation circuit is configured to receive a sequence of TIB elements and, for each TIB element received, produce a feed-forward value.

5. The systematic encoder apparatus of claim 4 , wherein the outer transform matrix is an IUT Toeplitz matrix, and

wherein the feed-forward computation circuit comprises a convolution circuit, the convolution circuit comprising a plurality of registers, multipliers, and adders, the convolution circuit being configured in accordance with an impulse response, the impulse response being derived from a first row of the IUT Toeplitz matrix.

6. The systematic encoder apparatus of claim 5 , wherein the outer encoder is further configured to use the feed-forward value together with a current element of the FDB or a current element of the OCB to generate a next element of the TIB and a next element of the TOB.

7. A method for use in a communication system for reliable transfer of a source data block (SDB), the method employing a systematic encoder apparatus configured for an outer transform matrix and an inner transform matrix, the method comprising:

configuring an inner encoder to receive the SDB and generate an output constraint block (OCB) as an image of the SDB under an inverse of a submatrix of the inner transform matrix; and

configuring an outer encoder to receive a fixed data block (FDB) and the OCB and generate a transform output block (TOB) subject to a first systematic coding constraint that the TOB is an image of a transform input block (TIB) under the outer transform matrix, a second systematic coding constraint that the TOB contains the OCB transparently in a sub-block of the TOB, and a third systematic coding constraint that the TIB contains the FDB transparently in a sub-block of the TIB,

wherein the inner encoder is further configured to receive the TOB and generate a transmitted code block (TCB) subject to a fourth systematic coding constraint that the TCB is an image of the TOB under the inner transform matrix and a fifth systematic coding constraint that the TCB contains the SDB transparently in a sub-block of the TCB.

8. The method of claim 7 , wherein the outer transform matrix is an invertible upper-triangular (IUT) matrix,

wherein the sub-block of the TOB comprises elements of the TOB whose indices belong to an index set, and

wherein the sub-block of the TIB comprises elements of the TIB whose indices belong to a complement of the index set.

9. The method of claim 8 , wherein the inner transform matrix is an invertible lower-triangular (ILT) matrix,

wherein the submatrix of the inner transform matrix comprises elements of the inner transform matrix whose row indices belong to the index set and whose column indices belong to the index set, and

wherein the sub-block of the TCB comprises elements of the TCB whose indices belong to the index set.

10. The method of claim 9 , wherein the outer encoder comprises a feed-forward computation circuit, and wherein the feed-forward computation circuit is configured to receive a sequence of TIB elements and, for each TIB element received, produce a feed-forward value.

11. The method of claim 10 , wherein the outer transform matrix is an IUT Toeplitz matrix, and

wherein the feed-forward computation circuit comprises a convolution circuit comprising a plurality of registers, multipliers, and adders, the convolution circuit configured in accordance with an impulse response, the impulse response being obtained from a first row of the IUT Toeplitz matrix.

12. The method of claim 11 , wherein the outer encoder is further configured to use the feed-forward value together with a current element of the FDB or a current element of the OCB to generate a next element of the TIB and a next element of the TOB.