大围长的缩短阵列码
Shortened Array Codes of Large Girth
摘要：
设计具有大围长的结构化低密度奇偶校验（LDPC）码的一种方法是缩短具有小围长的码，使得奇偶校验矩阵的删除列包含短循环中涉及的所有变量。如果码的奇偶校验矩阵是由循环置换矩阵的块组成的矩阵，则这种方法特别有效，如被称为阵列码的一类码的情况。我们展示了如何通过删除校验矩阵中的某些列来缩短阵列码，从而增加它们的围长。缩短方法基于以下观察：对于阵列码，并且实际上对于稍微更一般的LDPC码类，对应的Tanner图中的循环由具有整数系数的某些齐次线性方程支配。因此，我们可以选择性地从数组代码中消除循环，只保留原始代码的奇偶校验矩阵中由整数序列索引的那些列，这些整数序列不包含控制这些循环的方程的解。我们提供了可以从原始奇偶校验矩阵中保留的最大列数的拉姆齐理论估计，其属性是其索引序列避免求解各种类型的循环控制方程。这转化为缩短代码以消除周期所导致的速率损失的估计。仿真结果表明，对于所考虑的代码，缩短它们以增加围长可以导致显着的增益的信噪比的情况下，在加性高斯白噪声信道的通信。
英文摘要：
One approach to designing structured low-density parity-check (LDPC) codes with large girth is to shorten codes with small girth in such a manner that the deleted columns of the parity-check matrix contain all the variables involved in short cycles. This approach is especially effective if the parity-check matrix of a code is a matrix composed of blocks of circulant permutation matrices, as is the case for the class of codes known as array codes. We show how to shorten array codes by deleting certain columns of their parity-check matrices so as to increase their girth. The shortening approach is based on the observation that for array codes, and in fact for a slightly more general class of LDPC codes, the cycles in the corresponding Tanner graph are governed by certain homogeneous linear equations with integer coefficients. Consequently, we can selectively eliminate cycles from an array code by only retaining those columns from the parity-check matrix of the original code that are indexed by integer sequences that do not contain solutions to the equations governing those cycles. We provide Ramsey-theoretic estimates for the maximum number of columns that can be retained from the original parity-check matrix with the property that the sequence of their indices avoid solutions to various types of cycle-governing equations. This translates to estimates of the rate penalty incurred in shortening a code to eliminate cycles. Simulation results show that for the codes considered, shortening them to increase the girth can lead to significant gains in signal-to-noise ratio in the case of communication over an additive white Gaussian noise channel.
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