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Let m ≥ 2 be any natural number and let be a finite non-chain ring, where and q is a prime power congruent to 1 modulo (m-1). In this paper we study duadic codes over the ring and their extensions. A Gray map from to is defined which preserves self duality of linear codes. As a consequence self-dual, formally self-dual and self-orthogonal codes over are constructed. Some examples are also given to illustrate this.

Duadic codes form a class of cyclic codes that generalizes quadratic residue codes from prime to composite lengths. While initially quadratic residue codes were studied within the confines of finite fields, there have been recent developments on quadratic residue codes over some special rings. Pless and Qian [

and

extensions over the ring

There are duadic codes which are not quadratic residue codes, but they have properties similar to those of quadratic residue codes. In this paper we extend our

results of [

[

cases preserves self duality. The Gray images of extensions of duadic codes over the ring

The paper is organized as follows: In Section 2, we recall duadic codes of length n

over

In this section we give the definition of duadic codes and state some of their properties. Before that we need some preliminary notations and results.

A cyclic code

Let

idempotent

A polynomial

called odd like. A code

Suppose n is odd,

(i)

(iii) There exists a multiplier

Then codes

It is known that duadic codes exist if and only if q is a square mod n.

There is an equivalent definition of duadic codes in terms of idempotents. (For details see Huffman and Pless [

Let

(1) the idempotents satisfy

(2) There is a multiplier

i.e.

Associated to

If (1) and (2) hold we say that

Lemma 1: Let

(ii)

(iii)

(iv)

(v)

This is part of Theorem 6.1.3 of [

Lemma 2: Let

(i) If

then

(ii) If

then

Proof follows from Theorems 6.4.2 and 6.4.3 of [

Lemma 3:

Proof follows immediately from the definition and Lemma 1.

Let q be a prime power,

A simple calculation shows that

The decomposition theorem of ring theory tells us that

For a linear code

Then

product.

The following result is a simple generalization of a result of [

Theorem 1: Let

(i)

(ii) If

(iii) Further

(iv) Suppose that

(vi)

(vii)

The following is a well known result :

Lemma 4: (i) Let C be a cyclic code of length n over a finite ring S generated by the idempotent E in

(ii) Let C and D be cyclic codes of length n over a finite ring S generated by the idem-

potents

mpotents

Let the Gray map

where M is an

Let the Gray weight of an element

as

Theorem 2. The Gray map

distance preserving map from (

Further if the matrix V satisfies

of the matrix V, then the Gray image

dual code in

The proof follows exactly on the same lines as the proof of Theorem 2 of [

Proof. The first two assertions hold as

Let now

Let

implies that (comparing the coefficients of

for each r,

For convenience we call

Similarly

Using (2), we find that

Now

Using (3) and (4), one can check that each

We now define duadic codes over the ring

Lemma 5: Let

(1). Then for

potents not all equal and for any tuple

Throughout the paper we assume that q is a square mod n so that duadic codes of

length n over

For

In the same way, for

For

Similarly we define even-like idempotents for

Let

Theorem 3: Let

Proof: Let the multiplier

Note that

For a given positive integer k, the number of choices of the subsets

Let m be even first. Then

(16), we find that the number of inequivalent odd-like or even-like duadic-codes is

Let

Theorem 4: If

Proof: From the relations (2),(6)-(14) we see that

Using that

Similarly using

Therefore

Finally for

it being a repetition code over

This gives

since

Theorem 5 : If

(ii)

Proof: By using Lemma 2 and Lemma 4, we have

Similarly we get

Theorem 6 : If

The extended duadic codes over

as the extended duadic codes over

Consider the equation

This equation has a solution

Theorem 7: Suppose there exist a

Proof: As

where

ows from the fact that

Theorem 8: Suppose there exists a

Proof: Let

and

respectively where

vector of length n. As

rows of

are orthogonal to all the rows of

Corollary: Let the matrix V taken in the definition of the Gray map

Next we give some examples to illustrate our theory. The minimum distances of all the examples appearing have been computed by the Magma Computational Algebra System.

Example 1: Let

satisfying

Example 2: Let

be a matrix over

Example 3: Let

be a matrix over

Example 4 : Let

be a matrix over

Goyal, M. and Raka, M. (2016) Duadic Codes over the Ring and Their Gray Ima- ges. Journal of Computer and Communications, 4, 50-62. http://dx.doi.org/10.4236/jcc.2016.412003