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CPM-LDPC Codes Attaining the Minimum-Distance Bound

  • Kenta Kasai
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Abstract

We study binary quasi-cyclic LDPC codes whose parity-check matrices are full arrays of single circulant permutation matrices (CPMs), referred to here as CPM-LDPC codes. Their minimum distance is at most (J+1)!, where J is the column weight. For every fixed pair of column and row weights 2≤ J<L, we show that this bound is attained for all sufficiently large integer lift sizes. First, we give one integer exponent matrix independent of the lift size P. Second, we show that independent uniform exponent choices attain the bound with probability 1−OJ,L(P−1). Both proofs use cycle conditions required by low-weight codewords and a lower bound on the number of terms in vectors satisfying polynomial check equations. Neither construction requires P to be prime. We also give small-lift arrays attaining the bound 24 for J=3, L=4,…,8, and arrays with distance at least 28 for J=4, L=5,…,8, together with computational distance verification.

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Cite

BibTeXRIS
BibTeX
@misc{kasai2026cpmldpc,
  title = {{CPM-LDPC Codes Attaining the Minimum-Distance Bound}},
  author = {Kenta Kasai},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.16836},
  eprint = {2609.16836},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.16836},
}
RIS
TY  - UNPB
TI  - CPM-LDPC Codes Attaining the Minimum-Distance Bound
AU  - Kenta Kasai
PY  - 2026
DA  - 2026/09/15
PB  - arXiv
DO  - 10.48550/arxiv.2609.16836
UR  - https://arxiv.org/abs/2609.16836
AB  - We study binary quasi-cyclic LDPC codes whose parity-check matrices are full arrays of single circulant permutation matrices (CPMs), referred to here as CPM-LDPC codes. Their minimum distance is at most (J+1)!, where J is the column weight. For every fixed pair of column and row weights 2≤ J<L, we show that this bound is attained for all sufficiently large integer lift sizes. First, we give one integer exponent matrix independent of the lift size P. Second, we show that independent uniform exponent choices attain the bound with probability 1−O_J,L(P^−1). Both proofs use cycle conditions required by low-weight codewords and a lower bound on the number of terms in vectors satisfying polynomial check equations. Neither construction requires P to be prime. We also give small-lift arrays attaining the bound 24 for J=3, L=4,…,8, and arrays with distance at least 28 for J=4, L=5,…,8, together with computational distance verification.
ER  -