Abstract
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 nin {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 2ell -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 2ell -cycles. This is important for improving the performance of a code and avoiding the presence of low-weight codewords and absorbing sets.
Highlights
In the last three decades, the area of channel coding gained a lot of attention, due to the fact that many researchers were attracted by the practical realization of coding schemes whose performances approach the Shannon limit
We show how the parameters of the constructed convolutional code depend on the properties of the weak difference triangle set
We add some assumptions on the weak difference triangle sets (wDTSs) to be able to derive in this case a lower bound on the field size
Summary
In the last three decades, the area of channel coding gained a lot of attention, due to the fact that many researchers were attracted by the practical realization of coding schemes whose performances approach the Shannon limit. In [1], the authors constructed (n, n − 1)q LDPC convolutional codes, whose sliding parity-check matrix is free from 4 and 6-cycles not satisfying the so called full rank condition, starting from difference triangle sets This was a generalization of the work of Robinson and Bernstein, in which difference triangle sets were used to construct convolutional codes over the binary field, that can only avoid 4-cycles. In 1971, Tong [35] was the first to generalize their construction over Fq , using what we call in this paper weak difference triangle sets His construction is suitable only for limited rate and in a way that the Tanner graph associated to the parity-check matrix of these codes is free only from 4-cycles. We modify our construction to be able to relax these conditions on the wDTS, which in turn enlarges the underlying field size
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