Shortened Array Codes of Large Girth
Shortened Array Codes of Large Girth
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DOI：
10.1109/tit.2006.878179
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发表时间：
2005-04
影响因子：
2.5
通讯作者：
O. Milenkovic;N. Kashyap;David Leyba
中科院分区：
文献类型：
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作者：
O. Milenkovic;N. Kashyap;David Leyba
设计具有较大周长的结构性低密度平价检查（LDPC）代码的一种方法是缩短围栏较小的代码，以使均等检查矩阵的删除列包含涉及短周期中的所有变量。如果代码的奇偶校验检查矩阵是由循环置换矩阵块组成的矩阵，则这种方法特别有效，就像被称为阵列代码的代码类别一样。我们通过删除其奇偶校验检查矩阵的某些列来缩短阵列代码以增加周围的代码。缩短方法是基于以下观察结果：对于阵列代码，实际上，对于LDPC代码的一般类别，相应的Tanner图中的循环由具有整数系数的某些均匀的线性方程来控制。因此，我们可以通过仅保留原始代码的奇偶校验检查矩阵来选择性地消除数组代码中的周期，这些列是由整数序列索引的，而整数序列不包含对这些周期的方程式的解决方案。我们为最大的列数提供了Ramsey理论估计，可以从原始的奇偶校验检查矩阵中保留其属性的属性，该属性既可以避免使用解决各种类型的周期方程的解决方案。这意味着缩短代码以消除周期的费率罚款的估计值。仿真结果表明，对于所考虑的代码，缩短它们以增加周长可能会导致信噪比（SNR）的显着提高，而在加性白色高斯噪声（AWGN）频道上进行通信的情况下
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 (SNR) in the case of communication over an additive white Gaussian noise (AWGN) channel