Open Access Kent State (OAKS)

We investigate negacyclic and cyclic codes of length ps over the finite field Fpa. Negacyclic codes of length ps are precisely the ideals of the chain ring Fpa [x] / 〈xp^s+1〉. This structure is then used to obtain the Hamming distance distribution of the class of such negacyclic codes, which also provides Hamming weight distributions and enumerations of several codes. An one-to-one correspondence between negacyclic and cyclic codes is established to carry accordingly those results of negacyclic codes to cyclic codes.

For any prime p, all constacyclic codes of length ps over the ring ℛ = Fpm + uFpm are considered. The units of the ring ℛ are of the forms γ and ⍺ + uβ, where ⍺, β, and γ are nonzero elements of Fpm, which provides pm (pm -1) such constacyclic codes. First, the structure and Hamming distances of all constacyclic codes of length ps over the finite field Fpm are obtained; they are used as a tool to establish the structure and Hamming distances of all (⍺ + uβ)-constacyclic codes of length ps over ℛ. We then classify all cyclic codes of length ps over ℛ and obtain the number of codewords in each of those cyclic codes. Finally, a one-to-one correspondence between cyclic and γ-constacyclic codes of length ps over ℛ is constructed via ring isomorphism, which carries over the results regarding cyclic codes corresponding to γ-constacyclic codes of length ps over ℛ.

We study all constacyclic codes of length 2s over GR(Rfr,m), the Galois extension ring of dimension m of the ring Rfr=F2+uF2. The units of the ring GR(Rfr,m) are of the forms alpha, and alpha+ubeta, where alpha, beta are nonzero elements of F2m, which correspond to 2 m(2m-1) such constacyclic codes. First, the structure and Hamming distances of (1+ugamma)-constacyclic codes are established. We then classify all cyclic codes of length 2sover GR(Rfr,m), and obtain a formula for the number of those cyclic codes, as well as the number of codewords in each code. Finally, one-to-one correspondences between cyclic and alpha-constacyclic codes, as well as (1+ugamma)-constacyclic and (alpha+ubeta) -constacyclic codes are provided via ring isomorphisms, that allow us to carry over the results about cyclic and (1+ugamma)-constacyclic accordingly to all constacyclic codes of length 2s over GR(Rfr,m).

The units of the chain ring ℛa = Fpm [u]/〈ua〉 = Fpm + uFpm + ⋯ + ua−1Fpm are partitioned into a distinct types. It is shown that for any unit Λ of Type k, a unit λ of Type k∗ can be constructed, such that the class of λ-constacyclic of length ps of Type k∗ codes is one-to-one correspondent to the class of Λ-constacyclic codes of the same length of Type k via a ring isomorphism. The units of ℛa of the form Λ = Λ0 + u Λ1 + ⋯ + ua−1 Λa−1, where Λ0, Λ1, … , Λa−1 ∈ Fpm, Λ0 ≠ 0, Λ1 ≠ 0, are considered in detail. The structure, duals, Hamming and homogeneous distances of Λ-constacyclic codes of length ps over ℛa are established. It is shown that self-dual Λ-constacyclic codes of length ps over ℛa exist if and only if a is even, and in such case, it is unique. Among other results, we discuss some conditions when a code is both α- and β-constacyclic over ℛa for different units α, β.

Various kinds of distances of all negacyclic codes of length 2s over Zopf2a are completely determined. Using our structure theorems of negacyclic codes of length 2s over Zopf2a, we first calculate the Hamming distances of all such negacyclic codes, which particularly lead to the Hamming weight distributions and Hamming weight enumerators of several codes. These Hamming distances are then used to obtain their homogeneous, Lee, and Euclidean distances. Our techniques are extendable to the more general class of constacyclic codes, namely, the lambda- constacyclic codes of length 2s over Zopf2a, where lambda is any unit of Zopf2a with the form 4k-1. We establish the Hamming, homogeneous, Lee, and Euclidean distances of all such constacyclic codes.