We study constacyclic codes, of length $np^s$ and $2np^s$, that are generated by the polynomials $(x^n + \gamma)^{\ell}$ and $(x^n - \xi)^i(x^n + \xi)^j$\ respectively, where $x^n + \gamma$, $x^n - \xi$ and $x^n + \xi$ are irreducible over the alphabet $\F_{p^a}$. We generalize the results of [5], [6] and [7] by computing the minimum Hamming distance of these codes. As a particular case, we determine the minimum Hamming distance of cyclic and negacyclic codes, of length $2p^s$, over a finite field of characteristic $p$.
The minimum Hamming distance of cyclic codes, of length 2 s , over the Galois ring GR(2 a , m) is determined in [4]. In [5], the techniques introduced in [4] are used to compute the minimum Hamming distance of cyclic codes, of length p s , over a finite field of characteristic p.
It has been shown, in [2], that the minimum Hamming distance of a repeated root cyclic code can be expressed in terms of a simple root cyclic code. Using this result in [6], we have shown that the main result of [5] can be obtained immediately. More explicitly, we have shown that the minimum Hamming distance of a cyclic code, of length p s , over a finite field of characteristic p can be found using the results of [2] via simpler and more direct methods compared to those of [5]. Later in [7], we extended our methods, again using the results of [2], to cyclic codes, of length 2p s , over a finite field of characteristic p, where p is an odd prime, and we determined the minimum Hamming distance of these codes.
In this study, we generalize the results of [5], [6] and [7] to certain classes of repeated-root constacyclic codes. Namely, we compute the minimum Hamming distance of constacyclic codes of length np s and 2np s , that are generated by the polynomials (x n + λ) ℓ and (x nξ) i (x n +ξ) j respectively, where x n +λ, x n -ξ and x n +ξ are irreducible over the alphabet F p a . As a particular case, we determine the minimum Hamming distance of cyclic and negacyclic codes, of length 2p s , over a finite field of characteristic p.
This paper is organized as follows. In Section 2, we give some preliminaries and fix our notation. In Section 3, we determine the minimum Hamming distance of constacyclic codes, of length np s , over a finite field of characteristic p, where these code are generated by the irreducible polynomial x n + γ. In Section 4, we determine the minimum Hamming distance of constacyclic codes, of length 2np s , over a finite field of characteristic p, that are of the form (x n -ξ) i (x n + ξ) j where x n -ξ and x n + ξ are irreducible. In Section 5, we give several examples as applications of the main results of Section 3 and Section 4.
Let p be a prime number and F q be a finite field of characteristic p. Let N be a positive integer. Throughout this paper we identify a codeword c = (c 0 , c 1 , . . . , c N -1 ) over F q with the polynomial c(x
The Hamming weight of a codeword is defined to be the nonzero components of the codeword and the Hamming weight of a polynomial is defined to be the number of nonzero coefficients of the polynomial. Let c and c(x) be as above. We denote the Hamming weight of c and c(x) by w H (c) and w H (c(x)), respectively. Obviously, the Hamming weight of a codeword and the Hamming weight of the corresponding polynomial are equal, i.e., w H (c) = w H (c(x)).
The minimum Hamming distance of a code C is defined as min{w H (u -v) : u, v ∈ C and u = v}, and is denoted by d H (C). If C is a linear code, then it is well-known that
Let λ ∈ F q \ {0} and I = x N -λ . The λ-shift of a codeword c = (c 0 , c 1 , . . . , c N -1 ) is defined to be (λc N -1 , c 0 , c 1 , • • • , c N -2 ). If a linear code C is closed under λ-shifts, then C is called a λ-cyclic code and in general, such codes are called constacyclic codes (c.f. [1, Section 13.2]). It is well-known that λ-cyclic codes, of length N , over F q correspond to the ideals of the finite ring
In particular, cyclic (respectively negacyclic) codes, of length N , over F q correspond to the ideals of the ring
] is a principal ideal domain, R is also a principal ideal domain. So, for any ideal J of R, there exists a unique monic polynomial g(x) ∈ F q [x] with deg(g(x)) < N and g(x) | x N -λ such that J = g(x) . The polynomial g(x) is said to be a generator of J.
The following lemma gives us a trivial lower bound for the minimum Hamming distance of all constacyclic codes.
Then there is αx e ∈ C for some α ∈ F q \ {0} and for some nonnegative integer e. Since α and x are units, αx e is a unit in R. Being a proper ideal of R, C can not contain a unit. Thus we get a contradiction. Hence d H (C) ≥ 2. Now we will partition the set {1, 2, . . . , p s -1} into three subsets. These subsets naturally arise from the technicalities of our computations as described in Section 3 and Section 4. If i is an integer satisfying 1 ≤ i ≤ (p -1)p s-1 , then there exists a uniquely determined integer β such that 0 ≤ β ≤ p -2 and
for an integer i satisfying (p -1)p s-1 + 1 = p s -p s-1 + 1 ≤ i ≤ p s -1, there exists a uniquely determined integer k such that 1 ≤ k ≤ s -1 and
Besides if i is an integer as above and k is the integer satisfying 1 ≤ k ≤ s -1 and (2.1), then we have
. So for such integers i and k, there exists a uniquely determined integer τ with 1 ≤ τ ≤ p -1 such that
gives us a partition of the set {1, 2, . . . , p s -1}.
Here we fix some notation concerning division and remainders in
where either 0 ≤ deg(r(x)) < deg(f (x)) or r(x) = 0. We define f
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