The hulls of matrix-product codes over commutative rings and applications
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The hulls of matrix-product codes over commutative rings and applications Abdulaziz Deajim1
· Mohamed Bouye1 · Kenza Guenda2
Received: 7 June 2020 / Revised: 8 October 2020 / Accepted: 10 October 2020 © Korean Society for Informatics and Computational Applied Mathematics 2020
Abstract Given a commutative ring R with identity, a matrix A ∈ Ms×l (R), and linear codes C1 , . . . , Cs over R of the same length, this article considers the hull of the matrixproduct code [C1 . . . Cs ] A. Consequently, it introduces various sufficient conditions (as well as some necessary conditions in certain cases) under which [C1 . . . Cs ] A is a complementary dual (LCD) code. As an application, LCD matrix-product codes arising from torsion codes over finite chain rings are considered. Moreover, we show the existence of asymptotically good sequences of LCD matrix-product codes over such rings. Keywords Matrix-product codes · Hulls of codes · LCD codes · Torsion codes Mathematics Subject Classification 94B05 · 94B15 · 16S36
1 Introduction An active theme of research in coding theory is the construction of new codes by means of modifying or combining existing codes. In 2001, Blackmore and Norton [4] introduced the interesting and useful construction of matrix-product codes over finite fields. Such a construction included, as special cases, some previously well-known constructions such as the Plotkin’s (u|u + v)-construction, the (u + v + w|2u +
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Abdulaziz Deajim [email protected]; [email protected] Mohamed Bouye [email protected] Kenza Guenda [email protected]
1
Department of Mathematics, King Khalid University, P.O. Box 9004, Abha, Saudi Arabia
2
Laboratory of Algebra and Number Theory, Faculty of Mathematics, USTHB, BP 32, El Alia, Bab Ezzouar, Algeria
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v|u)-construction, the Turyn’s (a + x|b + x|a + b + x)-construction, and the (u + v|u−v)-construction. Through subsequent efforts of many researchers, matrix-product codes were further studied over finite fields and some types of finite commutative rings, see for instance [1,2,5,9,10,12,13], and [22]. In [19], J.L. Massey introduced the notion of linear complementary dual (LCD) codes over finite fields. Ever since, many subsequent papers on LCD codes and their applications over finite fields and some finite commutative rings have appeared, see for instance [7,11,15,16], and [23]. In a follow-up to [6], we consider here some aspects that connect the above tow notions: matrix-product codes and LCD codes over commutative rings. For this purpose, we first focus on studying the hull of matrix-product codes over such rings. We then use this to introduce various sufficient conditions under which a matrix-product code is an LCD code. We also give, in some special cases, necessary conditions for the construction of such codes. As an application, LCD matrix-product codes arising from torsion codes over finite chain rings are considered. Highlighting examples are also given. In order to put our results in a context as broad as possible, we assume in this ar
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