Notes on lcd codes over frobenius rings

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August undefined, 2024

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. 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.

Notes on LCD codes over Frobenius rings ScienceGate

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 … 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. … on shoes official

Frobenius Ring - an overview ScienceDirect Topics

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. 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 WebLCD code over a ﬁnite commutative ring were obtained. A linear code C with parameter [n,k,d] over a ﬁnite ﬁeld is said to be ... Remark 2.1. Note that, above inner product is generalization of Euclidean ... It is well known that for a Frobenius ring R and a linear code C over the ring R of length n, the product of cardinality of C and C ... iobroker matrix display

Non-Invertible-Element Constacyclic Codes over Finite PIRs

Cyclic codes over a non-chain ring R_e,q and their application to LCD …

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... WebJul 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. iobroker lovelace accweatherWebLCD 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 on shoes ombre

"WebKeywords Frobenius ring · Linear complementary dual code · Constacyclic code · Chain ring ... Recently, in [20], Liu and Liu studied LCD codes over ﬁnite chain rings and provided a necessary and sufﬁcient condition for a free linear code to be an LCD code over a ﬁnite chain ring. They also gave a sufﬁcient condition [20, Theorem 3.5 ... " - Notes on lcd codes over frobenius rings

Notes on lcd codes over frobenius rings

(PDF) Matrix product codes over finite commutative …

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 … 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 …

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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. 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

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 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 ...

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 ... 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.

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

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 … on shoes onwardWebMar 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 iobroker latest windows installerWebJan 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 … iobroker motioneyeWebWe 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 … on shoes order trackerWebJun 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. iobroker module not foundWebIn 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 … iobroker motioneye adapterWebOct 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; on shoes nz