Multiplicative Structure and Hecke Rings of Generator Matrices for Codes over Quotient Rings of Euclidean Domains

Abstract

Abstract:In this study, we consider codes over Euclidean domains modulo their ideals. In the first half of the study, we deal with arbitrary Euclidean domains. We show that the product of generator matrices of codes over the rings mod a and mod b produces generator matrices of all codes over the ring mod ab, i.e., this correspondence is onto. Moreover, we show that if a and b are coprime, then this correspondence is one-to-one, i.e., there exist unique codes over the rings mod a and mod b that produce any given code over the ring mod ab through the product of their generator matrices. In the second half of the study, we focus on the typical Euclidean domains such as the rational integer ring, one-variable polynomial rings, rings of Gaussian and Eisenstein integers, p-adic integer rings and rings of one-variable formal power series. We define the reduced generator matrices of codes over Euclidean domains modulo their ideals and show their uniqueness. Finally, we apply our theory of reduced generator matrices to the Hecke rings of matrices over these Euclidean domains.

A Modulus Factorization Algorithm for Self-Orthogonal and Self-Dual Quasi-Cyclic Codes via Polynomial Matrices

“…In other words, there exists reduced ∈ ( ) such that L = L . It is shown [4] that there exists one and only one reduced among for all ∈ ( ). From now on, unless otherwise noted, ∈ ( ) indicates the reduced generator matrix of an -module…”

“…i.e., { } is the set of the reduced generator matrices of all -modules in L/ L. Thus, we have the following one-to-one and onto correspondences [4]…”

“…Remark 1: By Proposition 1, an algorithm which computes is extracted as Algorithm 1 in [5]. If we estimate the computational complexity of Algorithm 1 as the total number of finite-field operations, it is evaluated approximately as 3 deg( ) , which is the same order as that of multiplying generator matrices in [4]. Proposition 2: (cf.…”

“…It is shown [3] that generator polynomial matrices are effective for the construction of quasi-cyclic codes which are equivalent to -modules in ( /(1 − ) ) , where = F [ ] and /(1 − ) is the quotient ring by an ideal (1 − ) in . In [4], codes over some rings are constructed by the product of generator polynomial matrices although it cannot efficiently construct self-orthogonal or self-dual codes because their product breaks these properties. In [5], selforthogonal and self-dual integer codes are constructed by using generator matrices and Chinese remainder theorem.…”

“…The rest of this paper is organized as follows. Section 2 prepares notations for -modules in ( / ) for some ∈ , where we refer to [3], [4], [5] for details. Section 3 describes the efficient construction of -modules in ( / ) , where we focus on the case of reciprocal , 1 , 2 for simplicity, and shows Example 1 of the above construction.…”