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Advances in Mathematics of Communications

November 2021 , Volume 15 , Issue 4

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Infinite families of $ 3 $-designs from o-polynomials
Cunsheng Ding and Chunming Tang
2021, 15(4): 557-573 doi: 10.3934/amc.2020082 +[Abstract](1523) +[HTML](829) +[PDF](381.17KB)
Abstract:

A classical approach to constructing combinatorial designs is the group action of a \begin{document}$ t $\end{document}-transitive or \begin{document}$ t $\end{document}-homogeneous permutation group on a base block, which yields a \begin{document}$ t $\end{document}-design in general. It is open how to use a \begin{document}$ t $\end{document}-transitive or \begin{document}$ t $\end{document}-homogeneous permutation group to construct a \begin{document}$ (t+1) $\end{document}-design in general. It is known that the general affine group \begin{document}$ {\mathrm{GA}}_1( {\mathrm{GF}}(q)) $\end{document} is doubly transitive on \begin{document}$ {\mathrm{GF}}(q) $\end{document}. The classical theorem says that the group action by \begin{document}$ {\mathrm{GA}}_1( {\mathrm{GF}}(q)) $\end{document} yields \begin{document}$ 2 $\end{document}-designs in general. The main objective of this paper is to construct \begin{document}$ 3 $\end{document}-designs with \begin{document}$ {\mathrm{GA}}_1( {\mathrm{GF}}(q)) $\end{document} and o-polynomials. O-polynomials (equivalently, hyperovals) were used to construct only \begin{document}$ 2 $\end{document}-designs in the literature. This paper presents for the first time infinite families of \begin{document}$ 3 $\end{document}-designs from o-polynomials (equivalently, hyperovals).

Rank weights for arbitrary finite field extensions
Grégory Berhuy, Jean Fasel and Odile Garotta
2021, 15(4): 575-587 doi: 10.3934/amc.2020083 +[Abstract](1383) +[HTML](641) +[PDF](364.62KB)
Abstract:

In this paper, we study several definitions of generalized rank weights for arbitrary finite extensions of fields. We prove that all these definitions coincide, generalizing known results for extensions of finite fields.

Involutory-Multiple-Lightweight MDS Matrices based on Cauchy-type Matrices
Mohsen Mousavi, Ali Zaghian and Morteza Esmaeili
2021, 15(4): 589-610 doi: 10.3934/amc.2020084 +[Abstract](1580) +[HTML](734) +[PDF](466.51KB)
Abstract:

One of the best methods for constructing maximum distance separable (\begin{document}$ \operatorname{MDS} $\end{document}) matrices is based on making use of Cauchy matrices. In this paper, by using some extensions of Cauchy matrices, we introduce several new forms of \begin{document}$ \operatorname{MDS} $\end{document} matrices over finite fields of characteristic 2. A known extension of a Cauchy matrix, called the Cauchy-like matrix, with application in coding theory was introduced in 1985. One of the main contributions of this paper is to apply Cauchy-like matrices to introduce \begin{document}$ 2n \times 2n $\end{document} involutory \begin{document}$ \operatorname{MDS} $\end{document} matrices in the semi-Hadamard form which is a generalization of the previously known methods. We make use of Cauchy-like matrices to construct multiple \begin{document}$ \operatorname{MDS} $\end{document} matrices which can be used in the Feistel structures. We also introduce a new extension of Cauchy matrices to be referred to as Cauchy-light matrices. The introduced Cauchy-light matrices are applied to construct \begin{document}$ n \times n $\end{document} \begin{document}$ \operatorname{MDS} $\end{document} matrices having at least \begin{document}$ 3n-3 $\end{document} entries equal to the unit element \begin{document}$ 1 $\end{document}; such a matrix is called a lightweight \begin{document}$ \operatorname{MDS} $\end{document} matrix and can be used in the lightweight cryptography. A simple closed-form expression is given for the determinant of Cauchy-light matrices.

An overview on skew constacyclic codes and their subclass of LCD codes
Ranya Djihad Boulanouar, Aicha Batoul and Delphine Boucher
2021, 15(4): 611-632 doi: 10.3934/amc.2020085 +[Abstract](1645) +[HTML](711) +[PDF](415.9KB)
Abstract:

This paper is about a first characterization of LCD skew constacyclic codes and some constructions of LCD skew cyclic and skew negacyclic codes over \begin{document}$ \mathbb{F}_{p^2} $\end{document}.

Quasi-symmetric designs on $ 56 $ points
Vedran Krčadinac and Renata Vlahović Kruc
2021, 15(4): 633-646 doi: 10.3934/amc.2020086 +[Abstract](1271) +[HTML](603) +[PDF](395.8KB)
Abstract:

Computational techniques for the construction of quasi-symmetric block designs are explored and applied to the case with \begin{document}$ 56 $\end{document} points. One new \begin{document}$ (56,16,18) $\end{document} and many new \begin{document}$ (56,16,6) $\end{document} designs are discovered, and non-existence of \begin{document}$ (56,12,9) $\end{document} and \begin{document}$ (56,20,19) $\end{document} designs with certain automorphism groups is proved. The number of known symmetric \begin{document}$ (78,22,6) $\end{document} designs is also significantly increased.

Correlation distribution of a sequence family generalizing some sequences of Trachtenberg
Ferruh Özbudak and Eda Tekin
2021, 15(4): 647-662 doi: 10.3934/amc.2020087 +[Abstract](1196) +[HTML](685) +[PDF](379.75KB)
Abstract:

In this paper, we give a classification of a sequence family, over arbitrary characteristic, adding linear trace terms to the function \begin{document}$ g(x) = \mathrm{Tr}(x^d) $\end{document}, where \begin{document}$ d = p^{2k}-p^k+1 $\end{document}, first introduced by Trachtenberg. The family has \begin{document}$ p^n+1 $\end{document} cyclically distinct sequences with period \begin{document}$ p^n-1 $\end{document}. We compute the exact correlation distribution of the function \begin{document}$ g(x) $\end{document} with linear \begin{document}$ m $\end{document}-sequences and amongst themselves. The cross-correlation values are obtained as \begin{document}$ C_{i,j}(\tau) \in \{-1,-1\pm p^{\frac{n+e}{2}},-1+p^n\} $\end{document}.

Infinite families of 2-designs from a class of non-binary Kasami cyclic codes
Rong Wang, Xiaoni Du and Cuiling Fan
2021, 15(4): 663-676 doi: 10.3934/amc.2020088 +[Abstract](1266) +[HTML](635) +[PDF](377.61KB)
Abstract:

Combinatorial \begin{document}$ t $\end{document}-designs have been an important research subject for many years, as they have wide applications in coding theory, cryptography, communications and statistics. The interplay between coding theory and \begin{document}$ t $\end{document}-designs has been attracted a lot of attention for both directions. It is well known that a linear code over any finite field can be derived from the incidence matrix of a \begin{document}$ t $\end{document}-design, meanwhile, that the supports of all codewords with a fixed weight in a code also may hold a \begin{document}$ t $\end{document}-design. In this paper, by determining the weight distribution of a class of linear codes derived from non-binary Kasami cyclic codes, we obtain infinite families of \begin{document}$ 2 $\end{document}-designs from the supports of all codewords with a fixed weight in these codes, and calculate their parameters explicitly.

Information set decoding in the Lee metric with applications to cryptography
Anna-Lena Horlemann-Trautmann and Violetta Weger
2021, 15(4): 677-699 doi: 10.3934/amc.2020089 +[Abstract](1387) +[HTML](595) +[PDF](451.62KB)
Abstract:

We convert Stern's information set decoding (ISD) algorithm to the ring \begin{document}$ \mathbb{Z}/4 \mathbb{Z} $\end{document} equipped with the Lee metric. Moreover, we set up the general framework for a McEliece and a Niederreiter cryptosystem over this ring. The complexity of the ISD algorithm determines the minimum key size in these cryptosystems for a given security level. We show that using Lee metric codes can substantially decrease the key size, compared to Hamming metric codes. In the end we explain how our results can be generalized to other Galois rings \begin{document}$ \mathbb{Z}/p^s\mathbb{Z} $\end{document}.

On finite length nonbinary sequences with large nonlinear complexity over the residue ring $ \mathbb{Z}_{m} $
Lin Yi, Xiangyong Zeng and Zhimin Sun
2021, 15(4): 701-720 doi: 10.3934/amc.2020091 +[Abstract](1120) +[HTML](579) +[PDF](443.91KB)
Abstract:

In this paper, we characterize all nonbinary sequences of length \begin{document}$ n $\end{document} with nonlinear complexity \begin{document}$ n-4 $\end{document} for \begin{document}$ n\geq9 $\end{document} and establish a formula on the number of such sequences. More generally, we characterize other finite length nonbinary sequences with large nonlinear complexity over \begin{document}$ \mathbb{Z}_{m} $\end{document}.

A generic construction of rotation symmetric bent functions
Junchao Zhou, Nian Li, Xiangyong Zeng and Yunge Xu
2021, 15(4): 721-736 doi: 10.3934/amc.2020092 +[Abstract](1104) +[HTML](557) +[PDF](428.05KB)
Abstract:

Rotation symmetric bent functions are a special class of Boolean functions, and their construction is of theoretical and practical interest. In this paper, we propose a generic construction of rotation symmetric bent functions by modifying the support of a known class of quadratic rotation symmetric bent functions, which generalizes some earlier works. Moreover, many infinite classes of rotation symmetric bent functions with maximal algebraic degree can be easily obtained from our construction.

$ \mathbb{Z}_{4}\mathbb{Z}_{4}[u] $-additive cyclic and constacyclic codes
Habibul Islam, Om Prakash and Patrick Solé
2021, 15(4): 737-755 doi: 10.3934/amc.2020094 +[Abstract](1304) +[HTML](673) +[PDF](437.82KB)
Abstract:

We study mixed alphabet cyclic and constacyclic codes over the two alphabets \begin{document}$ \mathbb{Z}_{4}, $\end{document} the ring of integers modulo \begin{document}$ 4 $\end{document}, and its quadratic extension \begin{document}$ \mathbb{Z}_{4}[u] = \mathbb{Z}_{4}+u\mathbb{Z}_{4}, u^{2} = 0. $\end{document} Their generator polynomials and minimal spanning sets are obtained. Further, under new Gray maps, we find cyclic, quasi-cyclic codes over \begin{document}$ \mathbb{Z}_{4} $\end{document} as the Gray images of both \begin{document}$ \lambda $\end{document}-constacyclic and skew \begin{document}$ \lambda $\end{document}-constacyclic codes over \begin{document}$ \mathbb{Z}_{4}[u] $\end{document}. Moreover, it is proved that the Gray images of \begin{document}$ \mathbb{Z}_{4}\mathbb{Z}_{4}[u] $\end{document}-additive constacyclic and skew \begin{document}$ \mathbb{Z}_{4}\mathbb{Z}_{4}[u] $\end{document}-additive constacyclic codes are generalized quasi-cyclic codes over \begin{document}$ \mathbb{Z}_{4} $\end{document}. Finally, several new quaternary linear codes are obtained from these cyclic and constacyclic codes.

Several new classes of (balanced) Boolean functions with few Walsh transform values
Tingting Pang, Nian Li, Li Zhang and Xiangyong Zeng
2021, 15(4): 757-775 doi: 10.3934/amc.2020095 +[Abstract](1308) +[HTML](564) +[PDF](399.29KB)
Abstract:

Three classes of (balanced) Boolean functions with few Walsh transform values derived from bent functions, Gold functions and the product of linearized polynomials are obtained in this paper. Further, the value distributions of their Walsh transform are also determined by virtue of the property of bent functions, the Walsh transform property of Gold functions and the \begin{document}$ k $\end{document}-tuple balance property of trace functions respectively.

2020 Impact Factor: 0.935
5 Year Impact Factor: 0.976
2020 CiteScore: 1.5

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