Mathematics

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14 pages, 288 KiB

Open AccessArticle

Twisted Hermitian Codes

by Austin Allen, Keller Blackwell, Olivia Fiol, Rutuja Kshirsagar, Bethany Matsick, Gretchen L. Matthews and Zoe Nelson

Mathematics 2021, 9(1), 40; https://doi.org/10.3390/math9010040 - 26 Dec 2020

Cited by 2 | Viewed by 2620

Abstract

We define a family of codes called twisted Hermitian codes, which are based on Hermitian codes and inspired by the twisted Reed–Solomon codes described by Beelen, Puchinger, and Nielsen. We demonstrate that these new codes can have high-dimensional Schur squares, and we identify [...] Read more.

We define a family of codes called twisted Hermitian codes, which are based on Hermitian codes and inspired by the twisted Reed–Solomon codes described by Beelen, Puchinger, and Nielsen. We demonstrate that these new codes can have high-dimensional Schur squares, and we identify a subfamily of twisted Hermitian codes that achieves a Schur square dimension close to that of a random linear code. Twisted Hermitian codes allow one to work over smaller alphabets than those based on Reed–Solomon codes of similar lengths. Full article

(This article belongs to the Special Issue Algebra and Its Applications)

19 pages, 315 KiB

Open AccessArticle

Consistent Flag Codes

by Clementa Alonso-González and Miguel Ángel Navarro-Pérez

Mathematics 2020, 8(12), 2234; https://doi.org/10.3390/math8122234 - 17 Dec 2020

Cited by 3 | Viewed by 2131

Abstract

In this paper we study flag codes on

F_{q}^{n}

, being

F_{q}

the finite field with q elements. Special attention is given to the connection between the parameters and properties of a flag code and the ones of a family [...] Read more.

In this paper we study flag codes on

F_{q}^{n}

, being

F_{q}

the finite field with q elements. Special attention is given to the connection between the parameters and properties of a flag code and the ones of a family of constant dimension codes naturally associated to it (the projected codes). More precisely, we focus on consistent flag codes, that is, flag codes whose distance and size are completely determined by their projected codes. We explore some aspects of this family of codes and present examples of them by generalizing the concepts of equidistant and sunflower subspace code to the flag codes setting. Finally, we present a decoding algorithm for consistent flag codes that fully exploits the consistency condition. Full article

(This article belongs to the Special Issue Algebra and Its Applications)

24 pages, 356 KiB

Open AccessArticle

New DNA Codes from Cyclic Codes over Mixed Alphabets

by Hai Q. Dinh, Sachin Pathak, Ashish Kumar Upadhyay and Woraphon Yamaka

Mathematics 2020, 8(11), 1977; https://doi.org/10.3390/math8111977 - 6 Nov 2020

Cited by 9 | Viewed by 2340

Abstract

Let

R = F_{4} + u F_{4}, with u^{2} = u

and [...] Read more.

Let

R = F_{4} + u F_{4}, with u^{2} = u

and

S = F_{4} + u F_{4} + v F_{4}, w i t h u^{2} = u, v^{2} = v, u v = v u = 0

. In this paper, we study

F_{4} R S

-cyclic codes of block length

(α, β, γ)

and construct cyclic DNA codes from them.

F_{4} R S

-cyclic codes can be viewed as

S [x]

-submodules of

\frac{F_{q} [x]}{⟨ x^{α} - 1 ⟩} \times \frac{R [x]}{⟨ x^{β} - 1 ⟩} \times \frac{S [x]}{⟨ x^{γ} - 1 ⟩}

. We discuss their generator polynomials as well as the structure of separable codes. Using the structure of separable codes, we study cyclic DNA codes. By using Gray maps

ψ_{1}

from R to

F_{4}^{2}

and

ψ_{2}

from S to

F_{4}^{3}

, we give a one-to-one correspondence between DNA codons of the alphabets

{A, T, G, C}^{2}, {A, T, G, C}^{3}

and the elements of

R, S

, respectively. Then we discuss necessary and sufficient conditions of cyclic codes over

F_{4}

, R, S and

F_{4} R S

to be reversible and reverse-complement. As applications, we provide examples of new cyclic DNA codes constructed by our results. Full article

(This article belongs to the Special Issue Algebra and Its Applications)

8 pages, 298 KiB

Open AccessArticle

Union of Sets of Lengths of Numerical Semigroups

by J. I. García-García, D. Marín-Aragón and A. Vigneron-Tenorio

Mathematics 2020, 8(10), 1789; https://doi.org/10.3390/math8101789 - 15 Oct 2020

Viewed by 1486

Abstract

Let

S = ⟨ a_{1}, \dots, a_{p} ⟩

be a numerical semigroup, let

s \in S

and let

Z (s)

be its set of factorizations. The set of lengths is denoted by [...] Read more.

Let

S = ⟨ a_{1}, \dots, a_{p} ⟩

be a numerical semigroup, let

s \in S

and let

Z (s)

be its set of factorizations. The set of lengths is denoted by

L (s) = {L (x_{1}, \dots, x_{p}) ∣ (x_{1}, \dots, x_{p}) \in Z (s)}

, where

L (x_{1}, \dots, x_{p}) = x_{1} + \dots + x_{p}

. The following sets can then be defined:

W (n) = {s \in S ∣ \exists x \in Z (s) such that L (x) = n}

,

ν (n) = ⋃_{s \in W (n)} L (s) = {l_{1} < l_{2} < \dots < l_{r}}

and

Δ ν (n) = {l_{2} - l_{1}, \dots, l_{r} - l_{r - 1}}

. In this paper, we prove that the function

Δ ν : N \to P (N)

is almost periodic with period

lcm (a_{1}, a_{p})

. Full article

(This article belongs to the Special Issue Algebra and Its Applications)

24 pages, 325 KiB

Open AccessArticle

Ore Extensions for the Sweedler’s Hopf Algebra H₄

by Shilin Yang and Yongfeng Zhang

Mathematics 2020, 8(8), 1293; https://doi.org/10.3390/math8081293 - 5 Aug 2020

Cited by 6 | Viewed by 1872

Abstract

The aim of this paper is to classify all Hopf algebra structures on the quotient of Ore extensions

H_{4} [z; σ]

of automorphism type for the Sweedler

^{'}

s 4-dimensional Hopf algebra

H_{4}

. Firstly, we calculate all [...] Read more.

The aim of this paper is to classify all Hopf algebra structures on the quotient of Ore extensions

H_{4} [z; σ]

of automorphism type for the Sweedler

^{'}

s 4-dimensional Hopf algebra

H_{4}

. Firstly, we calculate all equivalent classes of twisted homomorphisms

(σ, J)

for

H_{4}

. Then we give the classification of all bialgebra (Hopf algebra) structures on the quotients of

H_{4} [z; σ]

up to isomorphism. Full article

(This article belongs to the Special Issue Algebra and Its Applications)

5 pages, 183 KiB

Open AccessArticle

A New Secret Sharing Scheme Based on Polynomials over Finite Fields

by Selda Çalkavur, Patrick Solé and Alexis Bonnecaze

Mathematics 2020, 8(8), 1200; https://doi.org/10.3390/math8081200 - 22 Jul 2020

Cited by 6 | Viewed by 2964

Abstract

In this paper, we examine a secret sharing scheme based on polynomials over finite fields. In the presented scheme, the shares can be used for the reconstruction of the secret using polynomial multiplication. This scheme is both ideal and perfect. Full article

(This article belongs to the Special Issue Algebra and Its Applications)

8 pages, 256 KiB

Open AccessArticle

Estrada Index and Laplacian Estrada Index of Random Interdependent Graphs

by Yilun Shang

Mathematics 2020, 8(7), 1063; https://doi.org/10.3390/math8071063 - 1 Jul 2020

Cited by 9 | Viewed by 2709

Abstract

Let G be a simple graph of order n. The Estrada index and Laplacian Estrada index of G are defined by

E E (G) = \sum_{i = 1}^{n} e^{λ_{i} (A (G))}

and [...] Read more.

Let G be a simple graph of order n. The Estrada index and Laplacian Estrada index of G are defined by

E E (G) = \sum_{i = 1}^{n} e^{λ_{i} (A (G))}

and

L E E (G) = \sum_{i = 1}^{n} e^{λ_{i} (L (G))}

, where

{λ_{i} (A (G))}_{i = 1}^{n}

and

{λ_{i} (L (G))}_{i = 1}^{n}

are the eigenvalues of its adjacency and Laplacian matrices, respectively. In this paper, we establish almost sure upper bounds and lower bounds for random interdependent graph model, which is fairly general encompassing Erdös-Rényi random graph, random multipartite graph, and even stochastic block model. Our results unravel the non-triviality of interdependent edges between different constituting subgraphs in spectral property of interdependent graphs. Full article

(This article belongs to the Special Issue Algebra and Its Applications)

9 pages, 275 KiB

Open AccessArticle

The Square-Zero Basis of Matrix Lie Algebras

by Raúl Durán Díaz, Víctor Gayoso Martínez, Luis Hernández Encinas and Jaime Muñoz Masqué

Mathematics 2020, 8(6), 1032; https://doi.org/10.3390/math8061032 - 24 Jun 2020

Viewed by 2378

Abstract

A method is presented that allows one to compute the maximum number of functionally-independent invariant functions under the action of a linear algebraic group as long as its Lie algebra admits a basis of square-zero matrices even on a field of positive characteristic. [...] Read more.

A method is presented that allows one to compute the maximum number of functionally-independent invariant functions under the action of a linear algebraic group as long as its Lie algebra admits a basis of square-zero matrices even on a field of positive characteristic. The class of such Lie algebras is studied in the framework of the classical Lie algebras of arbitrary characteristic. Some examples and applications are also given. Full article

(This article belongs to the Special Issue Algebra and Its Applications)

14 pages, 272 KiB

Open AccessArticle

On Distance Signless Laplacian Spectral Radius and Distance Signless Laplacian Energy

by Luis Medina, Hans Nina and Macarena Trigo

Mathematics 2020, 8(5), 792; https://doi.org/10.3390/math8050792 - 14 May 2020

Cited by 8 | Viewed by 2353

Abstract

In this article, we find sharp lower bounds for the spectral radius of the distance signless Laplacian matrix of a simple undirected connected graph and we apply these results to obtain sharp upper bounds for the distance signless Laplacian energy graph. The graphs [...] Read more.

In this article, we find sharp lower bounds for the spectral radius of the distance signless Laplacian matrix of a simple undirected connected graph and we apply these results to obtain sharp upper bounds for the distance signless Laplacian energy graph. The graphs for which those bounds are attained are characterized. Full article

(This article belongs to the Special Issue Algebra and Its Applications)

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10 pages, 253 KiB

Open AccessArticle

On the Covering Radius of Codes over Z p k

by Mohan Cruz, Chinnapillai Durairajan and Patrick Solé

Mathematics 2020, 8(3), 328; https://doi.org/10.3390/math8030328 - 3 Mar 2020

Cited by 1 | Viewed by 1915

Abstract

In this correspondence, we investigate the covering radius of various types of repetition codes over

Z_{p^{k}} (k \geq 2)

with respect to the Lee distance. We determine the exact covering radius of the various repetition codes, which have been [...] Read more.

In this correspondence, we investigate the covering radius of various types of repetition codes over

Z_{p^{k}} (k \geq 2)

with respect to the Lee distance. We determine the exact covering radius of the various repetition codes, which have been constructed using the zero divisors and units in

Z_{p^{k}}

. We also derive the lower and upper bounds on the covering radius of block repetition codes over

Z_{p^{k}}

. Full article

(This article belongs to the Special Issue Algebra and Its Applications)

16 pages, 371 KiB

Open AccessArticle

On the (29, 5)-Arcs in PG(2, 7) and Some Generalized Arcs in PG(2, q)

by Iliya Bouyukliev, Eun Ju Cheon, Tatsuya Maruta and Tsukasa Okazaki

Mathematics 2020, 8(3), 320; https://doi.org/10.3390/math8030320 - 2 Mar 2020

Cited by 3 | Viewed by 2391

Abstract

Using an exhaustive computer search, we prove that the number of inequivalent

(29, 5)

-arcs in PG

(2, 7)

is exactly 22. This generalizes a result of Barlotti (see Barlotti, A. Some Topics in Finite Geometrical Structures, [...] Read more.

Using an exhaustive computer search, we prove that the number of inequivalent

(29, 5)

-arcs in PG

(2, 7)

is exactly 22. This generalizes a result of Barlotti (see Barlotti, A. Some Topics in Finite Geometrical Structures, 1965), who constructed the first such arc from a conic. Our classification result is based on the fact that arcs and linear codes are related, which enables us to apply an algorithm for classifying the associated linear codes instead. Related to this result, several infinite families of arcs and multiple blocking sets are constructed. Lastly, the relationship between these arcs and the Barlotti arc is explored using a construction that we call transitioning. Full article

(This article belongs to the Special Issue Algebra and Its Applications)

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10 pages, 233 KiB

Open AccessArticle

A Multisecret-Sharing Scheme Based on LCD Codes

by Adel Alahmadi, Alaa Altassan, Ahmad AlKenani, Selda Çalkavur, Hatoon Shoaib and Patrick Solé

Mathematics 2020, 8(2), 272; https://doi.org/10.3390/math8020272 - 18 Feb 2020

Cited by 21 | Viewed by 2803

Abstract

Secret sharing is one of the most important cryptographic protocols. Secret sharing schemes (SSS) have been created to that end. This protocol requires a dealer and several participants. The dealer divides the secret into several pieces ( the shares), and one share is [...] Read more.

Secret sharing is one of the most important cryptographic protocols. Secret sharing schemes (SSS) have been created to that end. This protocol requires a dealer and several participants. The dealer divides the secret into several pieces ( the shares), and one share is given to each participant. The secret can be recovered once a subset of the participants (a coalition) shares their information. In this paper, we present a new multisecret-sharing scheme inspired by Blakley’s method based on hyperplanes intersection but adapted to a coding theoretic situation. Unique recovery requires the use of linear complementary (LCD) codes, that is, codes in which intersection with their duals is trivial. For a given code length and dimension, our system allows dealing with larger secrets and more users than other code-based schemes. Full article

(This article belongs to the Special Issue Algebra and Its Applications)

13 pages, 263 KiB

Open AccessArticle

Projection Decoding of Some Binary Optimal Linear Codes of Lengths 36 and 40

by Lucky Galvez and Jon-Lark Kim

Mathematics 2020, 8(1), 15; https://doi.org/10.3390/math8010015 - 19 Dec 2019

Cited by 1 | Viewed by 2436

Abstract

Practically good error-correcting codes should have good parameters and efficient decoding algorithms. Some algebraically defined good codes, such as cyclic codes, Reed–Solomon codes, and Reed–Muller codes, have nice decoding algorithms. However, many optimal linear codes do not have an efficient decoding algorithm except [...] Read more.

Practically good error-correcting codes should have good parameters and efficient decoding algorithms. Some algebraically defined good codes, such as cyclic codes, Reed–Solomon codes, and Reed–Muller codes, have nice decoding algorithms. However, many optimal linear codes do not have an efficient decoding algorithm except for the general syndrome decoding which requires a lot of memory. Therefore, a natural question to ask is which optimal linear codes have an efficient decoding. We show that two binary optimal

[36, 19, 8]

linear codes and two binary optimal

[40, 22, 8]

codes have an efficient decoding algorithm. There was no known efficient decoding algorithm for the binary optimal

[36, 19, 8]

and

[40, 22, 8]

codes. We project them onto the much shorter length linear

[9, 5, 4]

and

[10, 6, 4]

codes over

G F (4)

, respectively. This decoding algorithm, called projection decoding, can correct errors of weight up to 3. These

[36, 19, 8]

and

[40, 22, 8]

codes respectively have more codewords than any optimal self-dual

[36, 18, 8]

and

[40, 20, 8]

codes for given length and minimum weight, implying that these codes are more practical. Full article

(This article belongs to the Special Issue Algebra and Its Applications)

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