Cyclotomic Constructions of Cyclic Codes with Length Being the Product of Two Primes

Cyclic codes are an interesting type of linear codes and have applications in communication and storage systems due to their efficient encoding and decoding algorithms. They have been studied for decades and a lot of progress has been made. In this paper, three types of generalized cyclotomy of order two and three classes of cyclic codes of length $n_1n_2$ and dimension $(n_1n_2+1)/2$ are presented and analysed, where $n_1$ and $n_2$ are two distinct primes. Bounds on their minimum odd-like weight are also proved. The three constructions produce the best cyclic codes in certain cases.Comment: 19 page

A Generalization of the Tang-Ding Binary Cyclic Codes

Cyclic codes are an interesting family of linear codes since they have efficient decoding algorithms and contain optimal codes as subfamilies. Constructing infinite families of cyclic codes with good parameters is important in both theory and practice. Recently, Tang and Ding [IEEE Trans. Inf. Theory, vol. 68, no. 12, pp. 7842--7849, 2022] proposed an infinite family of binary cyclic codes with good parameters. Shi et al. [arXiv:2309.12003v1, 2023] developed the binary Tang-Ding codes to the $4$-ary case. Inspired by these two works, we study $2^s$-ary Tang-Ding codes, where $s\geq 2$. Good lower bounds on the minimum distance of the $2^s$-ary Tang-Ding codes are presented. As a by-product, an infinite family of $2^s$-ary duadic codes with a square-root like lower bound is presented

Polyadic Constacyclic Codes

For any given positive integer $m$, a necessary and sufficient condition for the existence of Type I $m$-adic constacyclic codes is given. Further, for any given integer $s$, a necessary and sufficient condition for $s$ to be a multiplier of a Type I polyadic constacyclic code is given. As an application, some optimal codes from Type I polyadic constacyclic codes, including generalized Reed-Solomon codes and alternant MDS codes, are constructed.Comment: We provide complete solutions on two basic questions on polyadic constacyclic cdes, and construct some optimal codes from the polyadic constacyclic cde

Cyclic Codes from Cyclotomic Sequences of Order Four

Cyclic codes are an interesting subclass of linear codes and have been used in consumer electronics, data transmission technologies, broadcast systems, and computer applications due to their efficient encoding and decoding algorithms. In this paper, three cyclotomic sequences of order four are employed to construct a number of classes of cyclic codes over \gf(q) with prime length. Under certain conditions lower bounds on the minimum weight are developed. Some of the codes obtained are optimal or almost optimal. In general, the cyclic codes constructed in this paper are very good. Some of the cyclic codes obtained in this paper are closely related to almost difference sets and difference sets. As a byproduct, the $p$-rank of these (almost) difference sets are computed

Cyclotomy and duadic codes of prime lengths

10.1109/18.748995IEEE Transactions on Information Theory452453-466IETT