Binary quasi-reversible BCH codes whose defining set contains consecutive elements from negative to positive integers have received considerable attention in recent years due to their efficient decoding suitable for a wide range of applications. This paper combines the concept of the weight and quasi-reversible structures to introduce two subclasses of BCH codes: odd-like/even-like quasi-reversible BCH codes. The step-by-step decoding of these codes is developed as follows: First, the weight evaluation of a received polynomial is able to judge whether the number of errors is odd or even, which helps to simplify the decoding processes. Second, based on Chiò’s pivotal condensation process which can be easily implemented in a parallel computing architecture, the determinant calculation of the band matrix instead of Peterson’s matrix in column-echelon form is faster. Third, a newly proposed non-monic error-locator polynomial is sparser than the conventional ones. As a consequence, the theoretical analysis and experimental results validate potential benefits in requiring fewer finite field additions and multiplications used in the decoding of binary odd-like/even-like quasi-reversible BCH codes up to half the minimum distance when compared with the narrow-sense BCH codes with small error-correcting capability.

中文翻译：

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二进制准可逆 BCH 码的逐步解码


二进制准可逆BCH码的定义集包含从负整数到正整数的连续元素，由于其高效解码适合广泛的应用，近年来受到了广泛的关注。本文结合权值和准可逆结构的概念，介绍了BCH码的两个子类：类奇/类偶准可逆BCH码。这些代码的逐步解码发展如下：首先，对接收到的多项式进行权重评估，能够判断错误数量是奇数还是偶数，这有助于简化解码过程。其次，基于易于在并行计算架构中实现的Chiò关键凝聚过程，带状矩阵的行列式计算速度比列阶梯形式的Peterson矩阵更快。第三，新提出的非单调错误定位多项式比传统的多项式更稀疏。因此，理论分析和实验结果验证了与窄码相比，在二进制类奇/类偶准可逆 BCH 码解码中需要更少的有限域加法和乘法的潜在好处，最多可达最小距离的一半。 -检测具有小纠错能力的BCH码。