Abstract
Let p ≠ 3 be any prime. The structures of all λ-constacyclic codes of length 3p <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</sup> over the finite commutative chain ring Fp <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> + uFp <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> (u <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> = 0) are established in the term of their generator polynomials. As an application, Hamming and homogeneous distance of a class of such codes and RT distances of all are given. Among such λ-constacyclic codes, the unique maximum-distance-separable (briefly, MDS) code with respect to the RT distance is obtained. Moreover, when λ is not a cube in Fpm, the necessary and sufficient condition for the λ-constacyclic code of length 3p <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</sup> over Fp <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> + uFp <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> (u <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> = 0) be an MDS constacyclic code with respect to Hamming distance is provided.
Highlights
The class of constacyclic codes is an important class of linear codes in coding theory
Let λ be a unit in the finite field F, λ-constacyclic codes of length n over F are classified as the ideals g(x) of the ambient ring
In this article, we investigate the structure of λ-constacyclic codes of length 3ps over Fpm + uFpm, where p = 3 is a prime and λ is any unit in Fpm + uFpm
Summary
The class of constacyclic codes is an important class of linear codes in coding theory. In [3], Cao et al gave the structure of constacyclic codes of length nps over Fpm + uFpm. After that, some authors extended the study of codes over finite commutative rings to many more general lengths and alphabets (see, e.g., [18], [42], [49]). Constructing quantum error-correcting codes from all λ-constacyclic codes of length 3ps over Fpm + uFpm is an important direction for future work.
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