Notes on lcd codes over frobenius rings

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August undefined, 2024

WebLinear codes over a ﬁnite Frobenius ring also have the extension property [16]. Following a strategy of Dinh and L´opez-Permouth [4], the main result of this paper is that ﬁnite Frobenius rings are characterized by the extension property. That is, if the extension property holds for linear codes over a ﬁnite ring R,thenR is a WebNote that a left and right self-injective semi-primary ring is quasi-Frobenius. (4) The problem whether a self-injective cogenerator ring on the left side (left PF-ring) is a right PF-ring was unsolved until the middle of the 1980's (see Azumaya [Az66], Faith [F76]). Dischinger and W. MüUller [DM86] gave a counterexample in 1986 to this problem.

(PDF) Matrix product codes over finite commutative …

Weblinear ℓ-intersection pair of codes over a ﬁnite chain ring is observed as a generalization of LCD and LCP codes over a ﬁnite chain ring. LCD codes have been ﬁrst introduced by … WebJan 20, 2024 · 4 Constacyclic LCD codes over finite chain rings Lemma 7. For a polynomial f ( X) of degree r, f^* (X) denotes its reciprocal polynomial and is given by f^* (X)=X^rf... flower diaper

Notes on LCD Codes Over Frobenius Rings Semantic …

WebOct 6, 2024 · Notes on LCD Codes Over Frobenius Rings Abstract: Let R be a commutative Frobenius local ring. A result that the injective hull of an LCD code C over R. is free of dimension 1(C), where 1(C) is the minimum over the cardinalities of the generating sets … WebFeb 28, 2024 · The structure of the generalized Galois LCD codes over chain rings can be used to construct Galois LCD codes over finite fields. By using the structure of the … WebIn this paper, we clarify some aspects of LCD codes in the literature. We first prove that non-free LCD codes do not exist over finite commutative Frobenius local rings. We then obtain … greek vacations for couples

ON LCP AND CHECKABLE GROUP CODES OVER FINITE NON …

Galois LCD codes over rings - aimsciences.org

WebOct 6, 2024 · In this paper, we clarify some aspects of LCD codes in the literature. We first prove that non-free LCD codes do not exist over finite commutative Frobenius local rings. … WebWe define a basis of a code over a finite Frobenius ring as a system of vectors that is independent, modular independent, and spanning. As noted in [1, Remark 2], the two … greek vacations all inclusiveWebKeywords: Homogeneous weight, ﬁnite Frobenius rings, reﬂexive partitions MSC (2000): 94B05, 94B99, 16L60 1 Introduction The homogeneous weight has been studied extensively in the literature of codes over rings. It has been introduced by Constantinescu and Heise [6] as a generalization of both the Hamming flower diaper packing

"WebJan 30, 2024 · In this paper, we clarify some aspects on LCD codes in the literature. We first prove that a non-free LCD code does not exist over finite commutative Frobenius local rings. We then obtain a necessary and sufficient condition for the existence of LCD code over finite commutative Frobenius rings. " - Notes on lcd codes over frobenius rings

Notes on lcd codes over frobenius rings

WebOct 7, 2024 · We first prove that non-free LCD codes do not exist over finite commutative Frobenius local rings. We then obtain a necessary and sufficient condition for the existence of LCD codes over a finite ... WebMay 22, 2016 · It is proved that non-free LCD codes do not exist over finite commutative Frobenius local rings, and it is shown that a free constacyclic code over a finite chain ring is an LCD code if and only if it is reversible. 21 PDF ... 1 2 3 ... References SHOWING 1-6 OF 6 REFERENCES Hulls of cyclic and negacyclic codes over finite fields

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WebFrobenius ring, Section 3 will show its idempotents and the related decom-position of a R -module in RG. Sections 4 & 5 study when a group code is LCP in the case of the base ring being a a non-commutative ﬁnite local ring or a non-commutative ﬁnite Frobenius ring respectively. Finally, in Section 6, we deal with checkable codes in RG. 2. WebWe define a basis of a code over a finite Frobenius ring as a system of vectors that is independent, modular independent, and spanning. As noted in [1, Remark 2], the two properties of independence and modular independence are logically independent. Exercise 5.7 Let , and , . Show that the system is modular independent, but not independent.

WebNov 24, 2024 · 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. WebLCD code does not exist over ﬁnite commutative Frobenius local rings. We then obtain a necessary and sufﬁcient condition for the existence of LCD code over ﬁnite commutative Frobenius rings. We later show that a free constacyclic code over ﬁnite chain ring is LCD if and only if it is reversible, and

WebMar 31, 2024 · On LCP and checkable group codes over finite non-commutative Frobenius rings DeepAI On LCP and checkable group codes over finite non-commutative Frobenius rings 03/31/2024 ∙ by Sanjit Bhowmick, et al. ∙ Universidad de Valladolid ∙ Universidad Del Norte ∙ 0 ∙ share WebJul 5, 2024 · We give constructions of self-dual codes over any Frobenius ring. We describe connections to unimodular lattices, binary self-dual codes and to designs. We also describe linear complementary dual codes and make a new definition of a broad generalization encompassing both self-dual and linear complementary codes. Download chapter PDF

WebJan 30, 2024 · In this paper, we clarify some aspects on LCD codes in the literature. We first prove that a non-free LCD code does not exist over finite commutative Frobenius local …

WebJan 1, 2024 · Article on Galois LCD codes over rings, published in Advances in Mathematics of Communications 0 on 2024-01-01 by Zihui Liu. Read the article Galois LCD codes over rings on R Discovery, your go-to avenue for effective literature search. ... We describe the judging criterions for a generalized Galois LCD code over a Frobenius ring by introducing ... greek vanilla yogurt nutrition factsWebJan 30, 2024 · In Section 3, we discuss LCD codes over finite commutative Frobenius rings. Finally, Section 4 studies the complementary dual constacyclic codes over finite chain … flower dianthusWebOct 7, 2024 · Borello M de Cruz J Willems W A note on linear complementary pairs of group codes Discret. ... Fan Y Ling S Liu H Matrix product codes over finite commutative Frobenius rings Des. ... Liu XS Liu H LCD codes over finite chain rings Finite Field Appl. 2015 15 1 19 3333133 10.1016/j.ffa.2015.01.004 1392.94931 Google Scholar Digital Library; flower diceWebJul 24, 2024 · Some research work on LCD codes over finite Frobenius rings are also carried out. In 2015, Liu et al. [ 9] studied LCD codes over chain rings, and as the main result, they presented a condition for a free code over chain rings to be an LCD code. greek vase painting throw cushionWebFeb 1, 2024 · This paper sets a foundation for the study of linear codes over finite rings. The finite Frobenius rings are singled out as the most appropriate for coding theoretic … greek vacations packagesWebJun 15, 2024 · We define a Gray map φ and obtain many maximum-distance-separable (MDS) and optimal 𝔽_q-linear codes from the Gray images of cyclic codes. Under certain conditions we determine linear complementary dual (LCD) codes of length n when (n,q)≠ 1 and (n,q)= 1, respectively. greek vase painting cushionWebOct 1, 2024 · Existence of non-free LCD codes over R e, q. In [4], the authors proved that there does not exist any non-free LCD codes over a finite commutative local Frobenius ring, or in other words, every LCD code over a commutative local Frobenius ring is free. Also, the authors gave the conditions in case of a non-local ring for a code to be LCD. flower dibujo