Problem
Research questions and friction points this paper is trying to address.
CPM-LDPC Codes
Minimum Distance
Quasi-Cyclic LDPC
Circulant Permutation Matrices
CPM-LDPC Codes Attaining the Minimum-Distance Bound
🤖 AI Summary
研究了CPM-LDPC码，通过特定的整数指数矩阵和独立均匀指数选择方法，使得这些码达到最小距离界。
📝 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 $2\le J<L$, 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.
Innovation
Methods, ideas, or system contributions that make the work stand out.
CPM-LDPC Codes
Minimum-Distance Bound
Quasi-Cyclic LDPC Codes
Circulant Permutation Matrices
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