The Gray Image of Codes over Finite Chain Rings

Motivated by their work, we generalize their results to the case of (1 -γ e )constacyclic codes, cyclic codes and (1+γ e )-constacyclic codes over finite chain rings of any nilpotency index e + 1, where e ≥ 2.

We introduce a (1 -γ e )-constacyclic code and a (1 + γ e )-constacyclic code and characterize them in terms of corresponding polynomial representation in Section 2. In section 3, the Gray map on a finite chain ring R is defined and the Gray images of a (1 -γ e )-constacyclic code is investigated. The special case when the length of codes and the characteristic of the ring are relatively prime is treated in section 4.

A finite commutative ring with identity 1 = 0 is called a finite chain ring if its ideals are linearly ordered by inclusion. It is easily seen that a finite chain ring has a unique maximal ideal. Both the characteristic and the cardinality of a finite chain ring were shown, in [1], to be powers of the characteristic of its residue field. The nilpotency index of a finite chain ring is defined to be the smallest positive integer s such that γ s = 0 where γ is a generator of its maximal ideal. This γ plays a role of basis of the ring in the following sense: 4]). Let R be a finite chain ring of nilpotency index e + 1, γ a generator of its maximal ideal and {0} ⊆ V ⊆ R a set of representatives for the equivalence classes of R under congruence modulo γ. Assume that the residue field R/ γ is F p k where p is a prime. Then 1. for each r ∈ R there are unique a 0 (r), a 1 (r), . . . a e (r) ∈ V such that

As a consequence of Lemma 2.1, an element r ∈ R n can be written uniquely as

where a i (r) = (r i,0 , r i,1 , . . . , r i,n-1 ) ∈ V n , for each 0 ≤ i ≤ e. We denote a i (r) = ( r i,0 , r i,1 , . . . , r i,n-1 ) where : R → F n p k is the canonical map. A code of length n over a ring R is a nonempty subset of

Let R be a finite chain ring of nilpotency index e+1. Define ν : R n → R n by ν((r 0 , r 1 , . . . , r n-1 )) = ((1 -γ e )r n-1 , r 0 , . . . , r n-2 ).

A code

where

A linear cyclic code C over a ring is characterized in terms of corresponding polynomial representation,

Analogously, the polynomial representations of (1 -γ e )-constacyclic code and (1 + γ e )-constacyclic code are ideals of rings R

In this paper, we work over finite chain rings. Throughout, R denotes a finite chain ring of nilpotency index e + 1 where e ≥ 2 and the maximal ideal of R is generated by γ. The residue field R/ γ is regarded as F p k where p is a prime.

A homogeneous distance on R n is defined, in [2], in terms of the weight function w hom (r) defined as follows:

for all r = (r 0 , r 1 , . . . , r n-1 ) ∈ R n , where

The homogeneous distance d hom (r, s) between vectors r, s in R n is defined to be w hom (rs).

3 Gray Images of (1 -γ e )-constacyclic Codes

The Gray maps, which are defined in each case, have been used as tools to linked codes over rings and codes over finite fields. For a finite chain ring R, we define the Gay map from R n to F p ke n p k as the following:

In order to defined the Gray map from R n to F p ke n p k , recall that each element r ∈ R n can be written uniquely as

where a i (r) = (r i,0 , r i,1 , . . . , r i,n-1 ) ∈ V n , for every 0 ≤ i ≤ e.

Let α be a fixed primitive element of

where ξ i (ǫ) ∈ {0, 1, . . . , p -1}.

Likewise, each ω ∈ Z p ke is viewed uniquely as the p k -adic representation 1) , where ξ i (ω) ∈ {0, 1, . . . , p k -1}, for every 0 ≤ i ≤ e -1. Define the Gray map Φ :

for all 0 ≤ ω ≤ p k(e-1) -1 and 0 ≤ ǫ ≤ p k -1.

Notice that the Gray map Φ defined above is clearly injective. The distancepreserving property is shown in the next proposition. Proof. It suffices to show that w hom (r-s) = w H (Φ(r)-Φ(s)) for all r = s ∈ R, where w H the Hamming weight. We observe that Φ(rγ e ) = ( a 0 (r), a 0 (r), . . . , a 0 (r))

(2) Φ(r + sγ e ) = Φ(r) + Φ(sγ e ).

(3)

First, consider the case r -s ∈ Rγ e \ {0}. Then r -s = tγ e for some t ∈ R. It follows from (3) that Φ(r) -Φ(s) = Φ(tγ e ). Hence, by (2), w H (Φ(r) -Φ(s)) = w H (Φ(tγ e )) = p ke = w hom (r -s).

Next, assume that r -s ∈ R \ Rγ e . Write r = s + tγ m , where 0 ≤ m ≤ e -1 and t ∈ R\Rγ. In order to find w H (Φ(r)-Φ(s)), we need to count the number of 0 ≤ ω ≤ p k(e-1) -1 and 0

Since a i (tγ m ) = 0 for all 0 ≤ i ≤ m -1, we have a i (r) = a i (s) for all 0 ≤ i ≤ m -1 and the representative of a m (r) -a m (s) modulo γ is a m (tγ m ). It follows from a m (tγ m ) = 0 that a m (r) -a m (s) = 0. Consequently, equation ( 4) is a linear equation i