1. Introduction
Error correcting codes and error detection codes play an important role in data networks and satellite applications. Most coding theory is interested. Linear code has a clear structure and it is easy to find, understand, edit and decode for codes over finite rings. Since the 1970s, there are many research papers about codes over the finite ring. Several good nonlinear binary codes have been discovered. The circulation code on Z4 is composed of a Gary mapping structure [1]. After that, many researchers carried out more and more research on the code of finite ring [2] [3] [4] [5] [6]. The importance of finite rings in algebraic coding theory was established in the early 1990s by observing that some non-linear binary codes actually allow a linear representation of Z4 (see [1] [7]). It is also noted that the codes on the ring are particularly useful, if the distance function in the alphabet is not given by the usual Hamming metric, but by the homogeneous weight [8]. Examples of homogeneous weights are Hamming weights on finite fields and Lee weights on Z4. The homogeneous weight can be a natural extension of the Hamming weight of the code over finite rings.
In this paper, we will concern the linear code over the ring Z15, which has
elements and
. In the first section, we get the ideals of the ring Z15 through its non-units and give the Lee weights of elements in Z15. What’s more, we construct a unique expression of an element in Z15. In the second section, we obtain the generate matrix of the dual code of a linear code over the ring
, give the definition of Gray mapping from
to
and show that this Gray mapping is distance preserved. In the third section, we prove that the minimal Lee weight of C is equal to the minimal Hamming weight of its Gray Mapping. Further, the linear property of the Gray mapping of a linear code is obtained.
2. The Ring
The ring
is a non-chain ring, whose units are {1, 2, 4, 7, 8, 11, 13} and non-units are {0, 3, 5, 6, 9, 10, 12}. It has three ideals as follows,
,
,
The maximal ideal of the ring
are
and
, and we have
and
. By the Chinese Remainder Theorem, we have
.
Thus, for every
,
. Let
and
, then b and d in the expression are unique. The Lee weight of
is defined as
if
and
,
if
and
, and
if either
or
. Then the Lee weights of elements in
are as follows,
;
;
;
;
;
;
;
;
;
;
;
;
;
;
.
3. Linear Codes over the Ring
A linear code C over the ring
is a additive sub-module. For every code word
and
in C, the inner product of x and y is defined as
. x is orthogonal to y if
.
Let C be a linear code over the ring
with length n. The dual code of C is
. Thus,
is also a linear code over the ring
with length n. For a codeword
, the Lee weight of x is defined as
. For every two code words
, the Lee distance between x and y is
. The Hamming weight of x is
, and the Hamming distance between x and y is
.
By Chinese Reminder Theory, the generate matrix of the linear code C over the ring
is as follows
,
in which the elements of
and
belongs to
. Thus, C is a Abel Group of type
, and
.
Let H be the generate matrix of the dual code
. Then
, in which
is the rotate matrix of H. Since C is a Abel Group of type
, H is a Abel Group of type
. Let
By the linear transformation,
and
have can be changed to be the form as
According to
, we have
For every
, let
, in which
,
and
(
and
) for
. The Gray mapping from
to
is
It’s obvious that the Gray map is a distance preserved mapping from (
, Lee weight) to (
, Hamming weight).
4. Main Results
Theorem 1. For every
, we have
,
.
Therefore, the minimal Lee weight of C is equal to the minimal Hamming weight of
.
Proof: For
, let
, in which
and
. By the definition of Lee weight and Gray mapping, we have
,
Since
, we have
and
Thus we get the conclusion. □
Theorem 2. Let C be a linear code over the ring
with length n and d is the minimum distance over C. Then
is a line of code with the Parameter
.
Proof: For every
and
, let
,
. By the definition of gray mapping, we have
.
Thus
is linear. Since
,
is a linear code with the parameter
. □
The property of Theorem 2 also applies to the rings which can be decomposed into direct sum of two ideals.
5. Conclusion
By theoretical analysis and derivation, we prove that the minimum Lee weight of a linear code over the ring
is equal to the minimum Hamming weight of its Gray mapping. Furthermore, the linear property of its Gray mapping is confirmed. In this paper, we have done some preliminary research work, but there are also many other research contents about linear codes over the ring
to be considered, such as weight enumerators, Mac Williams identities and the self-dual codes over the ring
.
Acknowledgements
This research is supported by Foundation of Langfang Normal University (LSLB201707), Scientific Research Innovation Team of Langfang Normal University (Rings and Algebras with their applications on Error correcting theory), the Key Programs of Scientific Research Foundation of Hebei Educational Committee (Grant No.ZD2019056) and the Key Foundation of Hebei Education Department (ZD2017064).