Do non-free LCD codes over finite commutative Frobenius rings exist?

Bhowmick, Sanjit; Fotue-Tabue, Alexandre; Martı́nez-Moro, Edgar; Bandi, Ramakrishna; Bagchi, Satya

doi:10.1007/s10623-019-00713-x

Cited by 28 publications

(13 citation statements)

References 39 publications

(54 reference statements)

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“…Throughout this paper, we consider R as a commutative local ring. The following results are well known from [3].…”

Section: Preliminariesmentioning

confidence: 78%

Linear complementary dual code-based Multi-secret sharing scheme

Ghosh¹,

Bhowmick²,

Maurya³

et al. 2021

Preprint

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Hiding a secret is needed in many situations. Secret sharing plays an important role in protecting information from getting lost, stolen, or destroyed and has been applicable in recent years. A secret sharing scheme is a cryptographic protocol in which a dealer divides the secret into several pieces of share and one share is given to each participant. To recover the secret, the dealer requires a subset of participants called access structure. In this paper, we present a multi-secret sharing scheme over a local ring based on linear complementary dual codes using Blakley's method. We take a large secret space over a local ring that is greater than other code-based schemes and obtain a perfect and almost ideal scheme.Multi-secret sharing scheme (MSSS) is an important family of SSSs. It is a case in which many secrets need to be shared. In other words, a multi-secret sharing scheme is a protocol to share m arbitrarily related secrets s 1 , s 2 , . . . , s m among a

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“…Throughout this paper, we consider R as a commutative local ring. The following results are well known from [3].…”

Section: Preliminariesmentioning

confidence: 78%

Linear complementary dual code-based Multi-secret sharing scheme

Ghosh¹,

Bhowmick²,

Maurya³

et al. 2021

Preprint

Self Cite

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“…Let C = µ 1 g 1 (x) + µ 2 g 2 (x) be a cyclic code of length 10 over R 2,5 where g 1 (x) = x + 4 and g 2 (x) = (x + 1)(x + 4) 3 = x 4 + 3x 3 + 2x + 4. Therefore, ϕ(C) is a [20,15,4] linear code over F 5 and as per the database [16], it is an optimal linear code.…”

Section: Cyclic Codes Over R Eqmentioning

confidence: 99%

“…In [4], the authors proved that there does not exist any non-free LCD codes over finite commutative local Frobenius ring, or in other words, every LCD code over 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.…”

Section: Existence Of Non-free Lcd Codes Over R Eqmentioning

confidence: 99%

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Cyclic codes over a non-chain ring $R_{e,q}$ and their application to LCD codes

Islam,

Martínez-Moro,

Prakash

2021

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Let F q be a finite field of order q, a prime power integer such that q = et + 1 where t ≥ 1, e ≥ 2 are integers.In this paper, we study cyclic codes of length n over a non-chain ring R e,q = F q [u]/ u e −1 . 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 gcd(n, q) = 1 and gcd(n, q) = 1, respectively. It is proved that a cyclic code C of length n is an LCD code if and only if its Gray image ϕ(C) is an LCD code of length 4n over F q . Among others, we present the conditions for existence of free and non-free LCD codes. Moreover, we obtain many optimal LCD codes as the Gray images of non-free LCD codes over R e,q .

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“…Free cyclic serial codes have been determined by using cyclotomic cosets and trace map over finite chain rings [5]. It is clear that the Euclidean hull of cyclic codes is also cyclic, two special families of cyclic codes are of great interest, namely linear complementary dual codes, which are codes whose Euclidean hull is trivial (see for example [3]) and self-orthogonal codes, which are linear codes whose Euclidean hulls are the whole code (see for example [20,2]). These works motivate us to study the hulls of cyclic codes over finite chain rings.…”

Section: Introductionmentioning

confidence: 99%