Homogeneous metric and matrix product codes over finite commutative principal ideal rings

doi:10.1016/j.ffa.2020.101666

Finite Fields and Their Applications

Volume 64, June 2020, 101666

https://doi.org/10.1016/j.ffa.2020.101666 Get rights and content

Abstract

In this paper, a necessary and sufficient condition for the homogeneous distance on an arbitrary finite commutative principal ideal ring to be a metric is obtained. We completely characterize the lower bound of homogeneous distances of matrix product codes over any finite principal ideal ring where the homogeneous distance is a metric. Furthermore, the minimum homogeneous distances of the duals of such codes are also explicitly investigated.

Introduction

Constantinescu and Heise [4] first introduced the notion of homogeneous weight on the residue ring of integers. This weight is a generalization of the Hamming weight on finite fields and the Lee weight on the residue ring of integers modulo 4. This generalized weight is the only function that has the properties of homogeneity that are required to construct codes in coding theory. Later, Greferath and Schmidt [8] extended homogeneous weights to arbitrary finite rings, proved the existence and uniqueness of the homogeneous weight on finite rings, and exhibited a formula to calculate the homogeneous weight. However, it is not easy to determine the exact value of the homogeneous weight on finite rings. Fan and Liu [7] obtained an explicit formula of the homogeneous weight on finite principal ideal rings. In general, the homogeneous distance on a finite principal ideal ring induced by homogeneous weight may not be a metric. It is proved that [4], [12] the homogeneous distance on $Z_{N}$ is a metric if and only if N is not divisible by 6. There is no further result about when the homogeneous distance on an arbitrary finite commutative principal ideal ring is a metric.

Matrix product codes over finite fields were introduced in [1]. There are many such constructions, for example, the $(u | u + v)$ -construction and the $(a + x | b + x | a + b + x)$ -construction which can be viewed as matrix product codes. The reference [13] showed that some quasi-cyclic codes can also be rewritten as matrix product codes. The reference [1] also exhibited a lower bound for the minimum Hamming distances of matrix product codes over finite fields constructed by non-singular by columns matrices.

Codes over finite rings have received much attention recently after it was proved that some important families of binary non-linear codes are images under a Gray map of linear codes over $Z_{4}$ (see, for example, [2], [9], [15]). The references [17], [18] showed that only finite Frobenius rings are suitable for coding alphabets. In reference [6], the minimum Hamming distance of matrix product codes over finite Frobenius rings constructed with several types of matrices is bounded in different ways and the duals of matrix product codes are also explicitly described in terms of matrix product codes. The matrix product codes over some special finite Frobenius rings like finite chain rings and finite principal ideal rings have been investigated from many perspectives. In reference [16], matrix product codes over finite chain rings were studied and the lower bound on the minimum distance of matrix product codes by non-singular by columns matrices in [1] was extended to the minimum homogeneous distance. Let R be a finite commutative principal ideal ring with identity. Then $R = R_{1} \times R_{2} \times \dots \times R_{s}$ , where $R_{t}$ is a finite chain ring and $J_{t}$ is the unique maximal ideal of $R_{t}$ . Suppose $q_{t} = | R_{t} / J_{t} |$ for $1 \leq t \leq s$ , and $q_{1} \leq q_{2} \leq \dots \leq q_{s}$ . Reference [5] gave a lower bound for the minimum homogeneous distances of matrix product codes constructed by non-singular by columns matrices over the finite principal ideal ring R under the condition $q_{2} > q_{1} + 1$ provided $s \geq 2$ . Reference [5] also provided a lower bound for the minimum homogeneous distances of the duals of such codes constructed by square non-singular by columns matrices. Some other related results can be found in [3], [10], [11], [14].

In this paper, we first give a necessary and sufficient condition for the homogeneous distance on an arbitrary finite commutative principal ideal ring to be a metric. Then we completely characterize the lower bound of homogeneous distances of matrix product codes over any finite principal ideal ring where the homogeneous distance is a metric. As a consequence, the main results in reference [5] on the homogeneous distance lower bound when R satisfies $q_{2} > q_{1} + 1$ provided $s \geq 2$ are special cases of our results, from which they can be obtained directly. The minimum homogeneous distances of the duals of such codes constructed by arbitrary non-singular by columns matrices are also explicitly investigated. It generalizes the previous results in reference [5] where only square non-singular by columns matrices are considered.

This paper is organized as follows. Some necessary background materials are given in Section 2. We provide a necessary and sufficient condition for the homogeneous distance on a finite commutative principal ideal ring to be a metric in Section 3. Then we obtain a lower bound for the minimum homogeneous distances of matrix product codes and their duals constructed with non-singular by columns matrices over all such rings where the homogeneous distance is a metric in Section 4 and Section 5.

Section snippets

Finite commutative principal ideal rings and homogeneous weights

In this paper, R is always a finite commutative principal ideal ring with identity. Then R is isomorphic to a product of finite chain rings, which means that there exists a ring isomorphism $R \overset{≅}{⟶} R_{1} \times R_{2} \times \dots \times R_{s}, r ⟼ (r^{(1)}, r^{(2)}, \dots, r^{(s)}),$ where $R_{t}$ is a finite commutative chain ring with identity and $r^{(t)} \in R_{t}$ for $1 \leq t \leq s$ . With this isomorphism, we identify R with $R_{1} \times R_{2} \times \dots \times R_{s}$ and just write $r = (r^{(1)}, r^{(2)}, \dots, r^{(s)})$ in the following. For $t = 1, 2, \dots, s$ , let $J_{t}$ denote the unique maximal ideal of $R_{t}$ generated by $γ_{t}$ and $e_{t}$

Homogeneous distances

Recall that $R \overset{≅}{⟶} R_{1} \times R_{2} \times \dots \times R_{s}$ , where $R_{t}$ is a finite commutative chain ring with identity. For $t = 1, 2, \dots, s$ , $J_{t}$ is the unique maximal ideal of $R_{t}$ , $F_{t} : = R_{t} / J_{t}$ is a finite field and $q_{t} = | F_{t} |$ . Furthermore, we also assume that $q_{1} \leq q_{2} \leq \dots \leq q_{s}$ .

By Remark 2.1, the homogeneous distance $d_{h}$ on R is not always a metric on R. In this section, we provide a necessary and sufficient condition for the homogeneous distance $d_{h}$ on R to be a metric on R. It is easy to see that the homogeneous distance $d_{h}$ on R is a metric on R if

Homogeneous distance of matrix product codes

A lower bound for the minimum homogeneous distances of matrix product codes over finite principal ideal rings with $q_{2} > q_{1} + 1$ was obtained in the previous research [5]. In this section, we will completely characterize the lower bound of homogeneous distances of matrix product codes over any finite principal ideal ring.

A finite commutative principal ideal ring $R = R_{1} \times R_{2} \times \dots \times R_{s}$ can be divided into the following three types:

(T1)
$q_{1} = q_{2} = \dots = q_{s}$ , where $s \geq 1$ .
(T2)
$q_{t + 1} \geq q_{t} + 2$ provided $q_{1} = \dots = q_{t} < q_{t + 1} \leq q_{t + 2} \leq \dots \leq q_{s}$ , where $1 \leq t \leq s - 1$ .
(T3)
$q_{t +}$

Homogeneous distances of the duals

Theorem 4.1 gives a lower bound for the minimum homogeneous distances of the matrix product codes constructed with non-singular by columns matrices over R which satisfies (T1) or (T2). In this section, we will provide a lower bound for the minimum homogeneous distances of the duals of such codes. In particular, the main result in reference [5] on the lower bound of the dual codes of matrix product codes is a special case of our results. Note that the dual code of a matrix product code

Acknowledgements

This work was supported by NSFC (Grant No. 11871025).

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^☆: The main results in this paper have been reported at the 8th ICCM (June 2019) hosted by Yau Mathematical Sciences Center at Tsinghua University, Beijing, China.

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Homogeneous metric and matrix product codes over finite commutative principal ideal rings☆

Abstract

Introduction

Section snippets

Finite commutative principal ideal rings and homogeneous weights

Homogeneous distances

Homogeneous distance of matrix product codes

Homogeneous distances of the duals

Acknowledgements

Finite Fields Appl.

J. Comb. Theory, Ser. A

Quantum Inf. Process.

Matrix-product codes over Fq

Appl. Algebra Eng. Commun. Comput.

A linear construction for certain Kerdock and Preparata codes

Bull. Am. Math. Soc.

Matrix-product structure of constacyclic codes over finite chain rings Fpm[u]/〈ue〉

Appl. Algebra Eng. Commun. Comput.

A metric for codes over residue class rings of integers

Probl. Pereda. Inf.

Matrix product codes over finite commutative Frobenius rings

Des. Codes Cryptogr.

Homogeneous weights of finite rings and Möbius functions

Chin. Ann. Math., Ser. A

Matrix-product codes over $F_{q}$

Matrix-product structure of constacyclic codes over finite chain rings $F_{p^{m}} [u] / 〈 u^{e} 〉$