Linking Reversed and Dual Codes of Quasi-Cyclic Codes
Linking Reversed and Dual Codes of Quasi-Cyclic Codes
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DOI：
10.1587/transfun.2021tap0010
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发表时间：
2022
期刊：
影响因子：
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通讯作者：
R. T. Eldin;H. Matsui
中科院分区：
文献类型：
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作者：
R. T. Eldin;H. Matsui
摘要已知，与某些f𝑞[𝑥]模块相对应的有限型循环（QC）代码。 𝑥] - 模块。用R表示的C反向代码是通过C逆转C的所有代码，而C的双代码则用Cc⊥表示C c⊥如果在本研究中分别为r = c，c⊥=c⊥= c ，我们揭示了c，r和c⊥之间的关系，我们对c的条件对应于c⊥r，c = r = c⊥使用最佳参数的QC代码来表明存在各种可逆的自动QC代码，这些QC代码在线性代码的最小距离上实现了上限。
SUMMARY It is known that quasi-cyclic (QC) codes over the finite field F 𝑞 correspond to certain F 𝑞 [ 𝑥 ] -modules. A QC code C is specified by a generator polynomial matrix 𝐺 whose rows generate C as an F 𝑞 [ 𝑥 ] -module. The reversed code of C , denoted by R , is the code obtained by reversing all codewords of C while the dual code of C is denoted by C ⊥ . We call C reversible, self-orthogonal, and self-dual if R = C , C ⊥ ⊇ C , and C ⊥ = C , respectively. In this study, for a given C , we find an explicit formula for a generator polynomial matrix of R . A necessary and sufficient conditionfor C tobereversibleisderivedfromthisformula. Inaddition, we reveal the relations among C , R , and C ⊥ . Specifically, we give conditions on 𝐺 corresponding to C ⊥ ⊇ R , C ⊥ ⊆ R , and C = R = C ⊥ . As an application, we employ these theoretical results to the construction of QC codes with best parameters. Computer search is used to show that there exist various binary reversible self-orthogonal QC codes that achieve the upper bounds on the minimum distance of linear codes.