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In this paper, we study constacyclic codes of length n = 7 p s over a finite field of characteristics p , where p ≠ 7 is an odd prime number and s a positive integer. The previous methods in the literature that were used to compute the Hamming distances of repeated-root constacyclic codes of lengths n p s with 1 ≤ n ≤ 6 cannot be applied to completely determine the Hamming distances of those with n = 7. This is due to the high computational complexity involved and the large number of unexpected intermediate results that arise during the computation. To overcome this challenge, we propose a computer-assisted method for determining the Hamming distances of simple-root constacyclic codes of length 7, and then utilize it to derive the Hamming distances of the repeated-root constacyclic codes of length 7 p s. Our method is not only straightforward to implement but also efficient, making it applicable to these codes with larger values of n as well. In addition, all self-orthogonal, dual-containing, self-dual, MDS and AMDS codes among them will also be characterized. [ABSTRACT FROM AUTHOR]

Let p be an odd prime, and λ be a nonzero element of the finite field F p m . The λ -constacyclic codes of length 2 p s over F p m are classified as the ideals of quotient ring F p m [ x ] 〈 x 2 p s − λ 〉 in terms of their generator polynomials. Based on these generator polynomials, the symbol-pair distances of all such λ -constacyclic codes of length 2 p s are obtained in this paper. As an application, all MDS symbol-pair constacyclic codes of length 2 p s over F p m are established, which produce many new MDS symbol-pair codes with good parameters. [ABSTRACT FROM AUTHOR]

Abstract Let p be an odd prime, s , m be positive integers, γ , λ be nonzero elements of the finite field F p m such that γ p s = λ. In this paper, we show that, for any positive integer η , the Hamming distances of all repeated-root λ -constacyclic codes of length η p s can be determined by those of certain simple-root γ -constacyclic codes of length η. Using this result, Hamming distances of all constacyclic codes of length 4 p s are obtained. As an application, we identify all MDS λ -constacyclic codes of length 4 p s. [ABSTRACT FROM AUTHOR]

In this paper we introduce the notion of λ -constacyclic codes over finite rings R for arbitrary element λ of R. We study the non-invertible-element constacyclic codes (NIE-constacyclic codes) over finite principal ideal rings (PIRs). We determine the algebraic structures of all NIE-constacyclic codes over finite chain rings, give the unique form of the sets of the defining polynomials and obtain their minimum Hamming distances. A general form of the duals of NIE-constacyclic codes over finite chain rings is also provided. In particular, we give a necessary and sufficient condition for the dual of an NIE-constacyclic code to be an NIE-constacyclic code. Using the Chinese Remainder Theorem, we study the NIE-constacyclic codes over finite PIRs. Furthermore, we construct some optimal NIE-constacyclic codes over finite PIRs in the sense that they achieve the maximum possible minimum Hamming distances for some given lengths and cardinalities. [ABSTRACT FROM AUTHOR]

Let p ≠ 3 be a prime, s , m be positive integers, and λ be a nonzero element of the finite field F p m . In [22] and [20] , when the generator polynomials have one or two different irreducible factors, the Hamming distances of λ -constacyclic codes of length 3 p s over F p m have been considered. In this paper, we obtain that the Hamming distances of the repeated-root λ -constacyclic codes of length l p s can be determined by the Hamming distances of the simple-root γ -constacyclic codes of length l , where l is a positive integer and λ = γ p s . Based on this result, the Hamming distances of the repeated-root λ -constacyclic codes of length 3 p s are given when the generator polynomials have three different irreducible factors. Hence, the Hamming distances of all such constacyclic codes are determined. As an application, we obtain all optimal λ -constacyclic codes of length 3 p s with respect to the Griesmer bound and the Singleton bound. Among others, several examples show that some of our codes have the best known parameters with respect to the code tables in [19]. [ABSTRACT FROM AUTHOR]

Let p be a prime, s , m be positive integers, λ be a nonzero element of the finite field F p m . The b -distance generalizes the Hamming distance (b = 1) , and the symbol-pair distance (b = 2). While the Hamming and symbol-pair distances of all λ -constacyclic codes of length p s are completely determined, the general b -distance of such codes was left opened. In this paper, we provide a new technique to establish the b -distance of all λ -constacyclic codes of length p s , where 1 ≤ b ≤ ⌊ p 2 ⌋. As an application, all MDS b -symbol constacyclic codes of length p s over F p m are obtained. [ABSTRACT FROM AUTHOR]