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 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 …

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WebIn this work, we shall be largely concerned with codes over ﬁnite ﬁelds. However, we shall use codes over rings together with a linear Gray map to construct LCD codes. Hence we shall make the deﬁnitions in a general setting. A code C of length n over a ring R is a subset of Rn. All rings in this paper are assumed to be commutative 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 properties of independence and modular independence are logically independent. Exercise 5.7 Let , and , . Show that the system is modular independent, but not independent. solar of america

LCP of group codes over finite Frobenius rings - ResearchGate

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 … WebKeywords: 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 WebOct 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. solar offerte check

Frobenius Ring - an overview ScienceDirect Topics

Non-Invertible-Element Constacyclic Codes over Finite PIRs

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. WebFeb 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 … slurry supply systemWebIn 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 … slurry swivel

"WebJul 1, 2015 · Under suitable conditions, we provide a necessary and sufficient condition for LCD linear codes over finite chain rings. • We characterize LCD linear codes over finite … " - Notes on lcd codes over frobenius rings

Notes on lcd codes over frobenius rings

Do non-free LCD codes over finite commutative Frobenius …

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 … WebFeb 28, 2024 · We define the generalized Galois inner product for codes over Frobenius rings, and then present the Galois linear complementary dual (LCD) codes over such rings which generalize the Galois LCD codes over finite fields defined in a recent reference.

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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 … 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. …

WebJun 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. 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.

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... 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.

WebNotes on LCD codes over Frobenius rings IEEE Communications Letters . 10.1109/lcomm.2024.3029073 . 2024 . pp. 1-1. Author(s): Zihui Liu . Xin-Wen Wu. Keyword(s): Frobenius Rings . Lcd Codes Download Full-text Download PDF Package. Related Documents;

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 slurry surface treatmentWebJan 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 … slurry surfaceWebMay 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 solar off grid systemsWebDec 11, 2013 · Abstract. Properties of matrix product codes over finite commutative Frobenius rings are investigated. The minimum distance of matrix product codes … slurry supported excavationWebLCD 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 slurry submersible pumpWebNotes on LCD codes over Frobenius rings IEEE Communications Letters . 10.1109/lcomm.2024.3029073 . 2024 . pp. 1-1. Author(s): Zihui Liu . Xin-Wen Wu. … solaroll shutter repairsWebJan 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 ... solar-offerte-check