A method is proposed that utilizes punctured Reed-Solomon (RS) block codes for adaptive coding. Part of the redundancy of the RS codewords is used in a convolutional coding framework. This enables some codewords to use more redundancy for correcting errors, while other adjacent codewords use less redundancy.

On forward error correction with adaptive decoding (Corresp.)

Two convolutional-code construction schemes that utilize block codes are given. In the first method the generators of a self-orthogonal convolutional code (SOCC) are expanded. The generators of a block code whose block length is longer than that of the SOCC code replace the nonzero blocks of the convolutional code. The zero blocks are extended to the longer block length. There results a convolutional code whose blocks are self-orthogonal and which has a lower transmission rate. In the second scheme the parity constraints of an SOCC are expanded. The parity constraints of a block code replace some of the individual nonzero elements of the SOCC parity-check matrix, so that the convolutional code rate is greater than the block code rate. The resulting codes retain the SOCC advantages of simple implementation and limited error propagation. Both the encoding and the decoding can be based on the underlying block code. If a block code is majority decodable, then the resulting "hybrid" codes are majority decodable. Optimum majority-decodable block codes with up to five information bits per block are given, and from these codes several majority-decodable convolutional codes that are "optimum" with respect to the proposed construction are obtained.

Hybrid block- self-orthogonal convolutional codes

In this paper, a construction of $$(n,k,\delta )$$
 LDPC convolutional codes over arbitrary finite fields, which generalizes the work of Robinson and Bernstein and the later work of Tong is provided. The sets of integers forming a (k, w)-(weak) difference triangle set are used as supports of some columns of the sliding parity-check matrix of an $$(n,k,\delta )$$
 convolutional code, where $$n\in {\mathbb {N}}$$
 , $$n>k$$
 . The parameters of the convolutional code are related to the parameters of the underlying difference triangle set. In particular, a relation between the free distance of the code and w is established as well as a relation between the degree of the code and the scope of the difference triangle set. Moreover, we show that some conditions on the weak difference triangle set ensure that the Tanner graph associated to the sliding parity-check matrix of the convolutional code is free from $$2\ell $$
 -cycles not satisfying the full rank condition over any finite field. Finally, we relax these conditions and provide a lower bound on the field size, depending on the parity of $$\ell $$
 , that is sufficient to still avoid $$2\ell $$
 -cycles. This is important for improving the performance of a code and avoiding the presence of low-weight codewords and absorbing sets.

/pdf/construction-of-ldpc-convolutional-codes-via-difference-oifatr6amp.pdf

Construction of LDPC convolutional codes via difference triangle sets.

In this paper, a construction of $(n,k,\delta)$ LDPC convolutional codes over arbitrary finite fields, which generalizes the work of Robinson and Bernstein and the later work of Tong is provided. The sets of integers forming a $(k,w)$-(weak) difference triangle set are used as supports of some columns of the sliding parity-check matrix of an $(n,k,\delta)$ convolutional code, where $n\in\mathbb{N}$, $n>k$. The parameters of the convolutional code are related to the parameters of the underlying difference triangle set. In particular, a relation between the free distance of the code and $w$ is established as well as a relation between the degree of the code and the scope of the difference triangle set. Moreover, we show that some conditions on the weak difference triangle set ensure that the Tanner graph associated to the sliding parity-check matrix of the convolutional code is free from $2\ell$-cycles not satisfying the full rank condition over any finite field. Finally, we relax these conditions and provide a lower bound on the field size, depending on the parity of $\ell$, that is sufficient to still avoid $2\ell$-cycles. This is important for improving the performance of a code and avoiding the presence of low-weight codewords and absorbing sets.

Construction of LDPC convolutional codes via difference triangle sets

A class of convolutional, character-error-correcting codes with limited error propagation is presented. This class of codes is derived from binary convolutional self-orthogonal codes (BCSOC). By character-error-correcting, we mean that the code is character oriented, where each character can be thought of as a string of binary or higher base symbols of fixed length or as a single nonbinary symbol of correspondingly higher base. It is shown that, given a t -error-correcting BCSOC of rate b —1/ b , a character-error correcting convolutional self-orthogonal code (CCSOC) of rate k ( b —1)/( k ( b —1) + 1) can be constructed for any integer k , the rate expansion factor. The CCSOC so constructed corrects t character errors, and also possesses large simultaneous burst-error-correcting capabilities. Lower bounds on the burst-error-correcting capability for both BCSOC and CCSOC are found. Decoding consists of a mixture of majority logic decoding and algebraic computation. The decoding algorithm seems practical if either the rate expansion factor k or the number of errors corrected t are not large. Such codes are most suitable for channels with both random and burst noise, and also effect a compromise between the cost of terminal equipment and the efficient use of channels.

S.Y. Tong

Papers

Character-correcting convolutional self-orthogonal codes