Abstract
This paper presents a combinatorial construction of low-density parity-check (LDPC) codes from partially balanced incomplete block designs. Since Gallager’s construction of LDPC codes by randomly allocating bits in a sparse parity-check matrix, many researchers have used a variety of more structured combinatorial approaches. Many of these constructions start with the Galois field; however, this limits the choice of parameters of the constructed codes. Here we present a construction of LDPC codes of length \(4n^2 - 2n\) for all n using the cyclic group of order 2n. These codes achieve high information rate (greater than 0.8) for \(n \ge 8\), have girth at least 6 and have minimum distance 6 for n odd. The results provide proof of concept and lay the groundwork for potential high performing codes
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Notes
It is usual for the subset size to be denoted by k, but we use c instead, since in coding theory, k is normally reserved for the dimension of the code.
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Donovan, D., Price, A., Rao, A. et al. High-rate LDPC codes from partially balanced incomplete block designs. J Algebr Comb 55, 259–275 (2022). https://doi.org/10.1007/s10801-021-01111-0
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DOI: https://doi.org/10.1007/s10801-021-01111-0