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引用次数: 0
摘要
循环码作为一种特殊的常循环码，由于其良好的理论和数学性质而得到了广泛的研究。最近，Huang和Zhang （IEEE Trans Inf theory 70(4):2395 - 2410,2024）利用Mattson-Solomon多项式的导数研究了二进制扩展循环码的循环导数子代（DDs）和线性DDs，并提出了相应的导数译码方法。本文的目的之一是利用群代数理论将这些结论推广到q元扩展循环码。证明了q元扩展循环码的循环离散码是相同码，其线性离散码是等价码。此外，我们还证明了相关结果可以推广到q元常环码和由Plotkin构造生成的线性码。研究结果表明，Huang和Zhang提出的二进制循环码的软判决译码方法同样适用于q元循环码、q元常循环码和Plotkin构造生成的线性码。
Derivative descendants of cyclic codes and constacyclic codes
Cyclic codes, as a special type of constacyclic codes, have been extensively studied due to their favorable theoretical and mathematical properties. Very recently, by using the derivative of the Mattson-Solomon polynomials, Huang and Zhang (IEEE Trans Inf Theor 70(4):2395–2410, 2024) studied the cyclic derivative descendants (DDs) and linear DDs of binary extended cyclic codes and proposed the corresponding derivative decoding methods. One objective of this paper is to generalize these conclusions to q-ary extended cyclic codes with group algebra theory. It demonstrates that the cyclic DDs of a q-ary extended cyclic code are the same codes and its linear DDs are equivalent codes. In addition, we show that the relevant results can be generalized to q-ary constacyclic codes and the linear codes generated by Plotkin construction. Our conclusions reveal that the soft-decision decoding method proposed by Huang and Zhang for binary cyclic codes is also applicable to q-ary cyclic codes, q-ary constacyclic codes and the linear codes generated by Plotkin construction.
期刊介绍：
Designs, Codes and Cryptography is an archival peer-reviewed technical journal publishing original research papers in the designated areas. There is a great deal of activity in design theory, coding theory and cryptography, including a substantial amount of research which brings together more than one of the subjects. While many journals exist for each of the individual areas, few encourage the interaction of the disciplines.
The journal was founded to meet the needs of mathematicians, engineers and computer scientists working in these areas, whose interests extend beyond the bounds of any one of the individual disciplines. The journal provides a forum for high quality research in its three areas, with papers touching more than one of the areas especially welcome.
The journal also considers high quality submissions in the closely related areas of finite fields and finite geometries, which provide important tools for both the construction and the actual application of designs, codes and cryptographic systems. In particular, it includes (mostly theoretical) papers on computational aspects of finite fields. It also considers topics in sequence design, which frequently admit equivalent formulations in the journal’s main areas.
Designs, Codes and Cryptography is mathematically oriented, emphasizing the algebraic and geometric aspects of the areas it covers. The journal considers high quality papers of both a theoretical and a practical nature, provided they contain a substantial amount of mathematics.
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