(1+θ)-CONSTACYCLIC CODES OVER Z8 +θZ8

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. In this paper, (1+θ)-constacyclic codes of arbitrary length m over a non-chain finite local frobenious ring Z8 +θZ8 are introduced. A new Gray map is constructed from Z8 +θZ8 to Z8 and proved that the Z8 Gray image of (1+θ)-constacyclic codes having prescribed length m over the ring Z8 +θZ8 is a cyclic code of length 8m over the ring Z8. Moreover, it has been obtained that the binary image of the (1+θ)-constacyclic code of length m over Z8 +θZ8 is a distance invariant binary quasi-cyclic code of length 32m with index 16.


BACKGROUND
Many optimal binary linear codes have been studied from codes over several new classes of rings via some Gray map. Over the ring F 2 + uF 2 + vF 2 + uvF 2 , linear codes are discussed in [1], self dual codes in [2], cyclic codes in [3] and (1 + u)-constacyclic codes are described in [4] alongwith the construction of many optimal binary linear codes. More generally, cyclic codes over the ring R k were investigated in [12]. The rings mentioned above are not finite chain rings, however have rich algebraic structures and produce binary codes with large automorphism Now, defining the mappings ϖ, γ and ζ from R m to R m as follows: A unique set of generators for cyclic codes over Z 8 are discussed in the next lemma.
Lemma 2.2. Let C be a cyclic code of length m over Z 8 . Then, (2) If m is even, then are the binary polynomials with g(x)|(x m − 1)mod 2, and g(x)|p(x) (x m − 1) g(x) .
(ii): C = < g(x)+4p(x), 4a(x) >, where g(x), a(x) and p(x) are the binary polynomi- For a linear code C of length m over R, the two linear codes: Torsion code, Tor(C ) and Residue code, Res(C ) of length m over Z 8 are defined as: The homomorphism ϕ : R → Z 8 as ϕ(a + ϑ b) = a, extends naturally to a ring homomorphism Acting ϕ on C over R, define a ring homomorphism ϕ : By the application of first isomorphism theorem of finite groups, |C| = |Tor(C )||Res(C )|. Also, the image of C under the map ϕ is a cyclic code of length m over Z 8 .
Combining the above result with lemma 2.2, the set of generators for cyclic code of length m over R can be obtained as provided in following theorem. (1) If m is odd then, (2) If m is even then, , and for i = 1, 2, g i (x), a i (x) are the binary polynomial with g i (x)|(x m − 1)mod 2, and g i ( where g(x), a(x) and p(x) are the binary polynomials with a(x)|g( and deg(g(x))>deg(a(x))>deg(p(x)).

GRAY MAPS
Gray images of (1 + ϑ )-constacyclic codes over R The gray map ρ 1 from Z 8 to Z 4 2 defined as where z= p + 2q + 4r with p, q, r ∈ Z 2 , is a distance preserving map from Z m 8 (Lee distance) to Z 4m 2 (Hamming distance) and can be extended to Z m 8 as: Now, defining a new gray map ρ 2 from R m to Z 8n 8 as where c = a + ub and a, b ∈ Z 8 and can also be extended from R m to Z 8 as It is well known that the homogeneous weight has many applications for codes over finite rings and provides a good metric for the underlying ring in constructing superior codes. Next, a homogeneous weight on R is defined after defining of the homogeneous weight on arbitrary finite ring K .
(2) There exists a real number γ such that The Right homogeneous weight can be defined in a similar manner and if weight is both left homogeneous and right homogeneous, it is known as a homogeneous weight. For any element c By simple calculations the weight of any element c = a + ϑ b ∈ R is: It is easy to verify that, the above defined weight meets the conditions of the Definition 3.1, hence it is actually a homogeneous weight on R. The homogeneous distance of a linear code C over R, denoted by d hom (C ), is defined as the minimum homogeneous weight of the non-zero codewords of C .
The map ρ 2 is a distance preserving map from R m (homogeneous distance) to Z 8m 8 (Lee distance). Thus, we have the following three distance preserving maps: The following theorem defined a result on the above defined map ρ 2 .