Properties Of Structure And Period Distribution Of Some Cyclic Codes

Cyclic codes are an important sub-category of linear codes, invaluable in both theory and practice. Owing to their more rigorous algebraic structure over normal linear codes, cyclic codes are given particular attention by coding theorists and cryptographers. Cyclic codes also permit easier encoding and decoding in data transfer due to their properties.As theories for cyclic codes over finite fields mature, research has begun on cyclic codes over finite rings. Current literature mostly cover cyclic codes over the residual class ring Z 4and the four-element ring F₂ + uF₂, but there exists scarce literature on the eight-element ring F₂ + u²F₂ + u²F₂. This paper mainly discusses the structure and period distribution of cyclic codes over the eight-element ring F₂ + uF₂ + u F₂.The period distribution of cyclic codes is a novel idea in coding theory. At its core is a counting problem, which is a parameter of the code much like its quality distribution, code length and information rate. After the concept of period distribution was first proposed by Professors Yixian Yang and Zhengming Hu in 1992, many coding theorists and cryptographers have conducted research on, and yielded formulas for, period distribution of cyclic codes over finite fields Fq . By investigating the period distributions of codes, one can obtain better non-linear cyclic codes, as well as weight codes and permutation codes with stronger error-correction capabilities. There exist important practical uses of period distributions.The main contributions of this paper are as follows.(1) we discuss the factorization properties of polynomials with one variable over the non-unique factorization rings R + F₂ + u²F₂ + u²F₂. We prove that x n+ 1 yields the same basic polynomial factorization in R[ x ] and F₂ [ x ], disregarding associated elements, This forms the basis of research on cyclic codes over F₂ + u²F₂ + u F₂.(2) We discuss the structure of odd-length cyclic codes over F₂ + u²F₂ + u²F₂, and give the number of R-cyclic codes of code length n . This is the number of irreducible elements in the factorization of xⁿ+ 1.(3) We discuss the period distribution of cyclic codes over F₂ + u²F₂ + u²F₂, and give the formula for calculating periodic distributions of odd-length cyclic codes.