Constructions of block MDS LDPC codes from punctured circulant matrices

TL;DR

The paper addresses constructing block MDS LDPC codes that are also free of 4-cycles by leveraging punctured circulant matrices and their Moore/Vandermonde variants. It develops three algebraic frameworks: (i) punctured circulant permutation matrices (PUCPM) applicable over any finite field, (ii) punctured circulant matrices without puncturing in certain non-binary settings, and (iii) Moore-structured CM$(t)$ matrices with weight-2 blocks to enable long, high-rate binary codes. It provides Moore-determinant formulas and sufficient conditions to avoid short cycles, demonstrates binary and non-binary constructions (including several high-rate examples up to very long lengths), and compares favorably with existing LDPC codes and standards, with suggested applications to burst/randomburst channels and potential QKD use. The results advance practical block MDS LDPC code design by preserving algebraic structure while ensuring favorable Tanner-graph properties and strong error-correction performance. Future work points to richer Moore-type designs, improved masking strategies, and broader QKD-related applicability.

Abstract

Low density parity check (LDPC) codes, initially discovered by Gallager, exhibit excellent performance in iterative decoding, approaching the Shannon limit. MDS array codes, with favorable algebraic structures, are codes suitable for decoding large burst errors. The Blaum-Roth (BR) code, an MDS array code similar to the Reed-Solomon (RS) code but has a parity-check matrix prone to $4$-cycles. Fossorier proposed constructing quasi-cyclic LDPC codes from circulant permutation matrices but are not MDS array codes. This paper aims to construct codes that possess both the block MDS property and have no $4$-cycles in the Tanner graph of their parity-check matrices, namely the so-called block MDS LDPC codes. Non-binary block MDS QC codes were first constructed by [Tauz {\it et al. }IEEE ITW, 2025] using circulant shift matrices. We first generate a family of block MDS codes over $\F_2$ from punctured circulant permutation matrices. Second, we construct a family of block MDS LDPC codes from circulant matrices with column weight $> 1$ (CM$(t)$). Additionally, we present the Moore determinant formula for CM$(t)$s and a sufficient condition to avoid $4$-cycles in CM\((t)\)-QC LDPC codes' Tanner graphs for $t> 1$. We also point out the non-existence of binary block MDS CPM-QC LDPC codes. Compared to the codes constructed in [Li {\it et al. }IEEE TIT, 2023] and [Xiao {\it et al. }IEEE TCOM, 2021], our block MDS LDPC codes show enhanced random-error-correction at a similar code length and rate. Meanwhile, these codes can effectively combat burst errors when considered as array codes. Both of our two types of constructions for block MDS LDPC codes are applicable to the scenario of the binary field.

Figures (6)

• Figure 1: The BER and BLER performances of four LDPC codes
• Figure 2: The BER and BLER performances of eight LDPC codes
• Figure 3: The BER and BLER performances of ${\mathcal{C}}_9$ over $\mathbb{F}_{128}$
• Figure 4: The BER and BLER performances of five LDPC codes
• Figure 5: The BER and BLER performances of ${\mathcal{C}}_{11}$ and $B_{\text{PaG},Q}(4, 61)$.

Theorems & Definitions (26)

• Theorem 1
• Proposition 2
• Lemma 3
• Theorem 4
• Lemma 5
• Theorem 6
• proof
• Example 7
• Example 8
• Theorem 9