发表机构
School of Engineering, Institute of Science Tokyo（东京科学大学工程学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究二进制准循环LDPC码（CPM-LDPC码），证明其最小距离界$(J+1)!$对所有足够大的整数提升尺寸均可达到，并给出小提升阵列及计算验证。
AI 中文摘要
我们研究二进制准循环LDPC码，其奇偶校验矩阵是单个循环置换矩阵（CPM）的全阵列，这里称为CPM-LDPC码。它们的最小距离至多为$(J+1)!$，其中$J$是列重。对于每一对固定的列重和行重$2\le J<L$，我们证明对于所有足够大的整数提升尺寸，该界都能达到。首先，我们给出一个独立于提升尺寸$P$的整数指数矩阵。其次，我们证明独立的均匀指数选择以概率$1-O_{J,L}(P^{-1})$达到该界。两个证明都使用了低权重码字所需的循环条件和满足多项式校验方程的向量中项数的下界。两种构造都不要求$P$为素数。我们还给出了对于$J=3$，$L=4,\ldots,8$达到界24的小提升阵列，以及对于$J=4$，$L=5,\ldots,8$距离至少为28的阵列，并附有计算距离验证。
英文摘要
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 $2\le J<L$, 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 $J=3$, $L=4,\ldots,8$, and arrays with distance at least 28 for $J=4$, $L=5,\ldots,8$, together with computational distance verification.
Comments13 pages, 1 figure, 2 tables. Verification code and data: https://github.com/kasaikenta/cpm-ldpc-distance
