Recent zbMATH articles in MSC 94Bhttps://www.zbmath.org/atom/cc/94B2021-11-25T18:46:10.358925ZWerkzeugBook review of: E. Brown and R. K. Guy, The unity of combinatoricshttps://www.zbmath.org/1472.000102021-11-25T18:46:10.358925Z"Calkin, Neil"https://www.zbmath.org/authors/?q=ai:calkin.neil-j"Mulcahy, Colm"https://www.zbmath.org/authors/?q=ai:mulcahy.colmReview of [Zbl 1458.05001].Algebraic combinatorics. Translated from the Japanesehttps://www.zbmath.org/1472.050022021-11-25T18:46:10.358925Z"Bannai, Eiichi"https://www.zbmath.org/authors/?q=ai:bannai.eiichi"Bannai, Etsuko"https://www.zbmath.org/authors/?q=ai:bannai.etsuko"Ito, Tatsuro"https://www.zbmath.org/authors/?q=ai:ito.tatsuro"Tanaka, Rie"https://www.zbmath.org/authors/?q=ai:tanaka.rieThis book is the English translation of the book Introduction to Algebraic Combinatorics, which was published by Kyoritsu Shuppan in 2016 and written by the first three authors in Japanese. Later, the fourth author joined the team and translated the book in its current form in English. Bannai and Ito describe algebraic combinatorics as ``a group theory without groups'' or ``a character theoretical study of combinatorial objects''. The aim of this book is to ``pursue the study of combinatorics as an extension or a generalization of the study of finite permutation groups''. The early chapters of the book provides an accessible introduction to the subject for undergraduates and interested readers. The later chapters are suited for researchers in combinatorics and broader areas.
Chapter 1 is an introduction to classical combinatorics with selected subjects of study, such as graph theory and coding theory. Chapter 2 is an introduction to association schemes. Chapter 3 and 4 present codes and designs in connection to association schemes. Chapter 5 is on algebraic combinatorics on spheres. The book ends with Chapter 6 covering \(P\) and \(Q\)-polynomial schemes.Codes, cubes, and graphical designshttps://www.zbmath.org/1472.050302021-11-25T18:46:10.358925Z"Babecki, Catherine"https://www.zbmath.org/authors/?q=ai:babecki.catherineSummary: Graphical designs are an extension of spherical designs to functions on graphs. We connect linear codes to graphical designs on cube graphs, and show that the Hamming code in particular is a highly effective graphical design. We show that even in highly structured graphs, graphical designs are distinct from the related concepts of extremal designs, maximum stable sets in distance graphs, and \(t\)-designs on association schemes.On balanced \(( \mathbb{Z}_{4 u} \times \mathbb{Z}_{8 v}, \{4, 5 \}, 1)\) difference packingshttps://www.zbmath.org/1472.050322021-11-25T18:46:10.358925Z"Zhao, Hengming"https://www.zbmath.org/authors/?q=ai:zhao.hengming"Qin, Rongcun"https://www.zbmath.org/authors/?q=ai:qin.rongcun"Wu, Dianhua"https://www.zbmath.org/authors/?q=ai:wu.dianhuaSummary: Let \(K\) be a set of positive integers and let \(G\) be an additive group. A \((G, K, 1)\) difference packing is a set of subsets of \(G\) with sizes from \(K\) whose list of differences covers every element of \(G\) at most once. It is balanced if the number of blocks of size \(k \in K\) does not depend on \(k\). In this paper, we determine a balanced \(( \mathbb{Z}_{4 u} \times \mathbb{Z}_{8 v}, \{4, 5 \}, 1)\) difference packing of the largest possible size whenever \(uv\) is odd. The corresponding optimal balanced \((4 u, 8 v, \{4, 5 \}, 1)\) optical orthogonal signature pattern codes are also obtained.Möbius and coboundary polynomials for matroidshttps://www.zbmath.org/1472.051622021-11-25T18:46:10.358925Z"Johnsen, Trygve"https://www.zbmath.org/authors/?q=ai:johnsen.trygve"Verdure, Hugues"https://www.zbmath.org/authors/?q=ai:verdure.huguesSummary: We study how some coefficients of two-variable coboundary polynomials can be derived from Betti numbers of Stanley-Reisner rings. We also explain how the connection with these Stanley-Reisner rings forces the coefficients of the two-variable coboundary polynomials and Möbius polynomials to satisfy certain universal equations.Codes on linear sections of the Grassmannianhttps://www.zbmath.org/1472.140512021-11-25T18:46:10.358925Z"Carrillo-Pacheco, Jesús"https://www.zbmath.org/authors/?q=ai:carrillo-pacheco.jesus"Zaldívar, Felipe"https://www.zbmath.org/authors/?q=ai:zaldivar.felipeLet \(\mathbb{F}_q\) denote a finite field having \(q\) elements, where \(q\) is a prime power. For a vector space \(E\) of finite dimension \(m\) over \(\mathbb{F}_q\) and \(\ell\le m\), let \(G(\ell, m)\) denote the Grassmannian variety of vector subspaces of dimension \(\ell\) of \(E\). A projective variety \(X\) is called a linear section of the of the Grassmannian \(G(\ell,m)\) if \(X=G(\ell, m)\cap Z(g_1,\ldots, g_N)\), where \(g_1, g_2, \ldots, g_N\) are linearly independent functionals in the ideal that they generate and \(X(\mathbb{F}_q)=\{P_1, \ldots, P_M\}\) is a non-empty set of \(\mathbb{F}_q\)-rational points of \(X\). In this paper, authors study parity-check codes by showing that for every linear section of a Grassmannian, there exists a parity check code with good properties depending on the linear sections. For the Lagrangian-Grassmannian variety, they reveal that these parity-check codes are the low density parity check (LDPC) codes. They also obtained some properties of parity check codes associated to linear sections of Grassmannians.Estimation of m-sequences based on improved K-means algorithm and error correction of BCH codeshttps://www.zbmath.org/1472.621022021-11-25T18:46:10.358925Z"Qiang, Fang-Fang"https://www.zbmath.org/authors/?q=ai:qiang.fang-fang"Zhao, Zhi-Jin"https://www.zbmath.org/authors/?q=ai:zhao.zhijin"Chen, Ying"https://www.zbmath.org/authors/?q=ai:chen.ying.1|chen.ying.2|chen.ying"Shen, Lei"https://www.zbmath.org/authors/?q=ai:shen.lei"Xu, Chun-Yun"https://www.zbmath.org/authors/?q=ai:xu.chun-yunSummary: This study aimed to estimate multiple m-sequences and improve algorithm efficiency. For this, first, the existing improved K-means algorithm based on similarity measurement and silhouette coefficient theory was introduced. The Euclidean distance criterion was used to calculate the similarity to further improve the efficiency of the clustering algorithm. Then, the new improved K-means algorithm was used to estimate m-sequences and their number of types. Second, the equivalent Bose, Ray-Chaudhuri, and Hocquenghem (BCH) codes were constructed to correct the error bits in the m-sequences estimated using the clustering algorithm, thereby improving the estimation performance of m-sequences. The simulation results showed that compared with the existing algorithms, the performance of m-sequence estimation using the proposed algorithm was superior.Cryptanalysis of a code-based full-time signaturehttps://www.zbmath.org/1472.940672021-11-25T18:46:10.358925Z"Aragon, Nicolas"https://www.zbmath.org/authors/?q=ai:aragon.nicolas"Baldi, Marco"https://www.zbmath.org/authors/?q=ai:baldi.marco"Deneuville, Jean-Christophe"https://www.zbmath.org/authors/?q=ai:deneuville.jean-christophe"Khathuria, Karan"https://www.zbmath.org/authors/?q=ai:khathuria.karan"Persichetti, Edoardo"https://www.zbmath.org/authors/?q=ai:persichetti.edoardo"Santini, Paolo"https://www.zbmath.org/authors/?q=ai:santini.paolo-mariaSummary: We present an attack against a code-based signature scheme based on the Lyubashevsky protocol that was recently proposed by \textit{Y. Song} et al. [Theor. Comput. Sci. 835, 15--30 (2020; Zbl 1457.94222)]. The private key in the SHMWW scheme contains columns coming in part from an identity matrix and in part from a random matrix. The existence of two types of columns leads to a strong bias in the distribution of set bits in produced signatures. Our attack exploits such a bias to recover the private key from a bunch of collected signatures. We provide a theoretical analysis of the attack along with experimental evaluations, and we show that as few as 10 signatures are enough to be collected for successfully recovering the private key. As for previous attempts of adapting Lyubashevsky's protocol to the case of code-based cryptography, the SHMWW scheme is thus proved unable to provide acceptable security. This confirms that devising secure code-based signature schemes with efficiency comparable to that of other post-quantum solutions (e.g., based on lattices) is still a challenging task.New extremal Type II \(\mathbb Z_4\)-codes of length 32 obtained from Hadamard matriceshttps://www.zbmath.org/1472.940712021-11-25T18:46:10.358925Z"Ban, Sara"https://www.zbmath.org/authors/?q=ai:ban.sara"Crnković, Dean"https://www.zbmath.org/authors/?q=ai:crnkovic.dean"Mravić, Matteo"https://www.zbmath.org/authors/?q=ai:mravic.matteo"Rukavina, Sanja"https://www.zbmath.org/authors/?q=ai:rukavina.sanjaOn completely regular and completely transitive supplementary codeshttps://www.zbmath.org/1472.940722021-11-25T18:46:10.358925Z"Borges, J."https://www.zbmath.org/authors/?q=ai:borges.joaquim"Rifà, J."https://www.zbmath.org/authors/?q=ai:rifa.josep"Zinoviev, V. A."https://www.zbmath.org/authors/?q=ai:zinovev.viktor-aleksandrovichSummary: Given a parity-check matrix \(H_m\) of a \(q\)-ary Hamming code, we consider a partition of the columns into two subsets. Then, we consider the two codes that have these submatrices as parity-check matrices. We say that anyone of these two codes is the supplementary code of the other one.
We obtain that if one of these codes is a Hamming code, then the supplementary code is completely regular and completely transitive. If one of the codes is completely regular with covering radius 2, then the supplementary code is also completely regular with covering radius at most 2. Moreover, in this case, either both codes are completely transitive, or both are not.
With this technique, we obtain infinite families of completely regular and completely transitive codes which are quasi-perfect uniformly packed.A construction of \(\mathbb{F}_2 \)-linear cyclic, MDS codeshttps://www.zbmath.org/1472.940732021-11-25T18:46:10.358925Z"Cardell, Sara D."https://www.zbmath.org/authors/?q=ai:cardell.sara-d"Climent, Joan-Josep"https://www.zbmath.org/authors/?q=ai:climent.joan-josep"Panario, Daniel"https://www.zbmath.org/authors/?q=ai:panario.daniel"Stevens, Brett"https://www.zbmath.org/authors/?q=ai:stevens.brettIn this work, the authors introduce a family of codes of length \( n = p-1 \), for a prime \( p \), over the alphabet \( \mathbb{F}_2^b \), with the following properties: The codes are \( \mathbb{F}_2 \)-linear, the code redundancy over \( \mathbb{F}_2^b \), \( r \), satisfies \( n = rb \), the codes are cyclic over the alphabet \( \mathbb{F}_2^b \) (thus \( b \)-quasi-cyclic over \( \mathbb{F}_2 \)) and LDPC (with density going to \( 0 \) as \( n \longrightarrow \infty \)). Moreover, the codes are MDS over the alphabet \( \mathbb{F}_2^b \) in the case \( r = 2 \). The normalized dimension of the codes over the alphabet \( \mathbb{F}_2^b \) is \( k = n-r \) and satisfies \( k = r(b-1) \) since \( n = rb \). In other words, the information rate of the codes is \( k/n = 1 -1/b \), where \( b \) is the number of bits in a symbol of the code alphabet. MDS codes in such a scenario are of interest, for example, in Distributed Storage.
The codes are constructed using the concept of index array. They construct the parity-check matrices using such index arrays, which are in turn constructed using a partition of \( \mathbb{Z}_n \) and applying the Zech logarithm to the sets of the partition. The authors explicitly describe generator and parity-check matrices of the codes over \( \mathbb{F}_2 \). They then prove that the codes are MDS over the alphabet \( \mathbb{F}_2^b \) in the case \( r = 2 \) using graph-theoretic methods (including Hamiltonian paths). Finally, the authors provide an error-correcting algorithm for the given codes. However, the computational complexity of the algorithm is not explicitly given. Several examples are given throughout the article to illustrate the ideas.Error correcting code generated by induced subgraphs \(K_{m-1}\) of complete graph \(K_m\)https://www.zbmath.org/1472.940742021-11-25T18:46:10.358925Z"Choi, Sul-young"https://www.zbmath.org/authors/?q=ai:choi.sulyoungSummary: Consider the edge space of a complete graph on \(m\) vertices \(K_m\). Let \(\mathcal H\) be the subspace generated by the \(m\) induced subgraphs with \((m-1)\) vertices. If a vector \(z\) in \(\mathcal H\) is a sum of odd (or even) number of generating vectors of \(H\), \(z\) represents the edges of two disjoint complete subgraphs \(K_A\) and \(K_B\) (or the edges of the complete bipartite graph \(K_{A,B})\), where \(A\) is the intersection of vertices of the induced subgraphs corresponding to the generating vectors of \(z\) and \(B=V(K_m) - A\). We show that \(\mathcal H\) is an \((m(m-1)/2,m,m-1)\)-code when \(m\) is odd, and an \((m(m-1)/2, m-1,2(m-2))\)-code when \(m\) is even.New self-dual and formally self-dual codes from group ring constructionshttps://www.zbmath.org/1472.940752021-11-25T18:46:10.358925Z"Dougherty, Steven T."https://www.zbmath.org/authors/?q=ai:dougherty.steven-t"Gildea, Joe"https://www.zbmath.org/authors/?q=ai:gildea.joe"Kaya, Abidin"https://www.zbmath.org/authors/?q=ai:kaya.abidin"Yildiz, Bahattin"https://www.zbmath.org/authors/?q=ai:yildiz.bahattinSummary: In this work, we study construction methods for self-dual and formally self-dual codes from group rings, arising from the cyclic group, the dihedral group, the dicyclic group and the semi-dihedral group. Using these constructions over the rings \(\mathbb{F}_2+ u \mathbb{F}_2\) and \(\mathbb{F}_4+ u \mathbb{F}_4,\) we obtain 9 new extremal binary self-dual codes of length 68 and 25 even formally self-dual codes with parameters \([72,36,14]\).Extended maximal self-orthogonal codeshttps://www.zbmath.org/1472.940762021-11-25T18:46:10.358925Z"Durğun, Yilmaz"https://www.zbmath.org/authors/?q=ai:durgun.yilmazOn polycyclic codes over a finite chain ringhttps://www.zbmath.org/1472.940772021-11-25T18:46:10.358925Z"Fotue-Tabue, Alexandre"https://www.zbmath.org/authors/?q=ai:tabue.alexandre-fotue"Martínez-Moro, Edgar"https://www.zbmath.org/authors/?q=ai:martinez-moro.edgar"Blackford, J. Thomas"https://www.zbmath.org/authors/?q=ai:blackford.jason-thomasThis paper explore polycyclic codes over finite chain rings. For a finite chain ring \(S\), these codes are identified with ideals of the quotient ring \(S[X]/\left\langle X^n- a(X) \right\rangle,\) where \(a(X)\) is a polynomial in \(S[X]\) such that the constant term be a unit. The intersection and sum of free polycyclic codes over \(S\) are characterized and the structure of non-free polycyclic codes is described by using strong Gröbner bases. It is shown that an \(S\)-linear code is polycyclic if and only if its dual is a sequential code. It is proved that the annihilator code and the euclidean orthogonal have the same type. The fact that the annihilator dual of polycyclic code (respectively free-polycyclic code) is also polycyclic code (respectively free-polycyclic code) is established. It is shown that the Galois image of non-free non-zero polycyclic codes is polycyclic and a Gröbner basis is provided. Necessary and sufficient condition for a free polycyclic code to be Galois-disjoint (respectively complete Galois disjoint) is given. The trace codes and restrictions of free polycyclic codes are characterized giving an analogue of Delsarte's theorem relating the trace code and the annihilator dual code.Some remarks on non projective Frobenius algebras and linear codeshttps://www.zbmath.org/1472.940782021-11-25T18:46:10.358925Z"Gómez-Torrecillas, José"https://www.zbmath.org/authors/?q=ai:gomez-torrecillas.jose"Hieta-Aho, Erik"https://www.zbmath.org/authors/?q=ai:hieta-aho.erik"Lobillo, F. J."https://www.zbmath.org/authors/?q=ai:lobillo.francisco-javier"López-Permouth, Sergio"https://www.zbmath.org/authors/?q=ai:lopez-permouth.sergio-r"Navarro, Gabriel"https://www.zbmath.org/authors/?q=ai:navarro.gabriel.2In this paper, the notion of non projective Frobenius algebras and their application to linear codes are developed. The usual definition of a Frobenius algebra is extended by removing the projectivity technical requirement: ``an algebra \(R\) over a commutative Frobenius ring \(K\) is said to be non projective Frobenius if it is finitely generated as \(K\)-module and there exists a certain non degenerate \(K\)-bilinear form.''
It is shown that an \(R\)-algebra over a commutative Frobenius ring \(K\), which is also finitely generated as a \(K\)-module is a Frobenius ring if and only if \(R\) is a non-projective Frobenius \(K\)-algebra. Some well-known equivalent conditions used to characterize finite Frobenius ring are extended and the generating character is generalized by introducing the notion of Frobenius functional. It is illustrated that a Frobenius \(K\)-algebra is non projective Frobenius but a non projective Frobenius algebra need not be projective over its commutative base Frobenius ring. A method for constructing new finite Frobenius rings from skew polynomial rings is described.
Some main results of [\textit{S. Szabo} and \textit{J. A. Wood}, J. Algebra Comb. Discrete Struct. Appl. 4, No. 2, 105--113 (2017; Zbl 1423.94156)] are analyzed. Results on annihilators associated to a non degenerate bilinear form from a finite Frobenius ring to a non projective algebra are extended and it is deduced that a non projective algebra over a Frobenius commutative ring is a quasi-Frobenius ring. Bilinear forms defined on modules over non-projective Frobenius algebras are described and are used to develop a general version of MacWilliams identity.New doubly even self-dual codes having minimum weight 20https://www.zbmath.org/1472.940792021-11-25T18:46:10.358925Z"Harada, Masaaki"https://www.zbmath.org/authors/?q=ai:harada.masaakiSummary: In this note, we construct new doubly even self-dual codes having minimum weight 20 for lengths 112, 120 and 128. This implies that there are at least three inequivalent extremal doubly even self-dual codes of length 112.A complete classification of partial MDS (maximally recoverable) codes with one global parityhttps://www.zbmath.org/1472.940802021-11-25T18:46:10.358925Z"Horlemann-Trautmann, Anna-Lena"https://www.zbmath.org/authors/?q=ai:horlemann-trautmann.anna-lena"Neri, Alessandro"https://www.zbmath.org/authors/?q=ai:neri.alessandroSummary: We generalize the definition of partial MDS codes to locality blocks of various length and show that these codes are maximally recoverable. Then we focus on partial MDS codes with exactly one global parity. We derive a general construction for such codes by describing a suitable parity check matrix. Then we give a construction of generator matrices of such codes. Afterwards we show that all partial MDS codes with one global parity have a generator matrix (or parity check matrix) of this form. This gives a complete classification and hence also a sufficient and necessary condition on the underlying field size for the existence of such codes. This condition is related to the classical MDS conjecture. Moreover, we investigate the decoding of such codes and give some comments on partial MDS codes with more than one global parity.New extremal binary self-dual codes from a Baumert-Hall arrayhttps://www.zbmath.org/1472.940812021-11-25T18:46:10.358925Z"Kaya, Abidin"https://www.zbmath.org/authors/?q=ai:kaya.abidin"Yildiz, Bahattin"https://www.zbmath.org/authors/?q=ai:yildiz.bahattinSummary: In this work, we introduce new construction methods for self-dual codes using a Baumert-Hall array. We apply the constructions over the alphabets \(\mathbb{F}_2\) and \(\mathbb{F}_4 + u \mathbb{F}_4\) and combine them with extension theorems and neighboring constructions. As a result, we construct 46 new extremal binary self-dual codes of length 68, 26 new best known Type II codes of length 72 and 8 new extremal Type II codes of length 80 that lead to new \(3 -(80, 16, 665)\) designs. Among the new codes of length 68 are the examples of codes with the rare \(\gamma = 5\) parameter in \(W_{68, 2} \). All these new codes are tabulated in the paper.Systematic and nonsystematic perfect codes of infinite length over finite fieldshttps://www.zbmath.org/1472.940822021-11-25T18:46:10.358925Z"Malyugin, Sergeĭ Artem'evich"https://www.zbmath.org/authors/?q=ai:malyugin.sergey-artemevichSummary: Let \(F_q\) be a finite field of \(q\) elements (\(q = p^k\), \(p\) is a prime number). An infinite-dimensional \(q\)-ary vector space \(F_q^{\mathbb{N}_0}\) consists of all sequences \(u = (u_1, u_2, \dots)\), where \(u_i \in F_q\) and all \(u_i\) are 0 except some finite set of indices \(i \in \mathbb{N}\). A subset \(C \subset F_q^{\mathbb{N}_0}\) is called a perfect \(q\)-ary code with distance 3 if all balls of radius 1 (in the Hamming metric) with centers in \(C\) are pairwise disjoint and their union covers the space. Define the infinite perfect \(q\)-ary Hamming code \(H_q^\infty\) as the infinite union of the sequence of finite \(q\)-ary codes \(\tilde{H}_q^n\) where for all \(n = (q^m -1)/(q -1)\), \(\tilde{H}_q^n\) is a subcode of \(\tilde{H}_q^{qn+1}\). We prove that all linear perfect \(q\)-ary codes of infinite length are affine equivalent. A perfect \(q\)-ary code \(C \subset F_q^{\mathbb{N}_0}\) is called systematic if \(\mathbb{N}\) could be split into two subsets \(N_1\), \(N_2\) such that \(C\) is a graphic of some function \(f : F_q^{N_{0,1}} \rightarrow F_q^{N_{2,0}}\). Otherwise, \(C\) is called nonsystematic. Further general properties of systematic codes are proved.
We also prove a version of Shapiro-Slotnik theorem for codes of infinite length. Then, we construct nonsystematic codes of infinite length using the switchings of \(s < q - 1\) disjoint components. We say that a perfect code \(C\) has the complete system of triples if for any three indices \(i_1\), \(i_2\), \(i_3\) the set \(C- C\) contains the vector with support \(\{i_1, i_2, i_3\}\). We construct perfect codes of infinite length having the complete system of triples (in particular, such codes are nonsystematic). These codes can be obtained from the Hamming code \(H_q^\infty\) by switching some family of disjoint components \(\mathcal{B} = \{R^{u_1}_1, R^{u_2}_2, \dots\}\). Unlike the codes of finite length, the family \(\mathcal{B}\) must obey the rigid condition of sparsity. It is shown particularly that if the family of componentsB does not satisfy the condition of sparsity then it can generate a perfect code having noncomplete system of triples.Complete weight enumerators of two classes of linear codes with a few weightshttps://www.zbmath.org/1472.940832021-11-25T18:46:10.358925Z"Qi, Minglong"https://www.zbmath.org/authors/?q=ai:qi.minglong"Xiong, Shengwu"https://www.zbmath.org/authors/?q=ai:xiong.shengwuAuthors' abstract: Linear codes with a few weights have important applications in secret sharing, authentication codes, data storage system, association schemes, and strongly regular graphs. Hence, the construction of linear codes with a few weights is an important research topic in coding theory. In this paper, we construct two new classes of linear codes with two and three weights, and determine their complete weight enumerators. Our work generalizes the results of [\textit{Q. Wang} et al., Discrete Math. 340, No. 3, 467--480 (2017; Zbl 1407.94183)].On the non-existence of linear perfect Lee codes: the Zhang-Ge condition and a new polynomial criterionhttps://www.zbmath.org/1472.940842021-11-25T18:46:10.358925Z"Qureshi, Claudio"https://www.zbmath.org/authors/?q=ai:qureshi.claudio-mSummary: The Golomb-Welch conjecture [\textit{S. W. Golomb} and \textit{L. R. Welch}, SIAM J. Appl. Math. 18, 302--317 (1970; Zbl 0192.56302)] states that there are no \(e\)-perfect Lee codes in \(\mathbb{Z}^n\) for \(n \geq 3\) and \(e \geq 2\). This conjecture remains open even for linear codes. A recent result of Zhang and Ge establishes the non-existence of linear \(e\)-perfect Lee codes in \(\mathbb{Z}^n\) for infinitely many dimensions \(n,\) for \(e = 3\) and 4. In this paper we extend this result in two ways. First, using the non-existence criterion of \textit{T. Zhang} and \textit{G. Ge} [IEEE Trans. Inf. Theory 63, No. 7, 4325--4331 (2017; Zbl 1370.94580)] together with a generalized version of Lucas' theorem we extend the above result for almost all \(e \) (i.e. a subset of positive integers with density 1). Namely, if \(e\) contains a digit 1 in its base-3 representation which is not in the unit place (e.g.\( e = 3, 4)\) there are no linear \(e\)-perfect Lee codes in \(\mathbb{Z}^n\) for infinitely many dimensions \(n\). Next, based on a family of polynomials (the \(Q\)-polynomials), we present a new criterion for the non-existence of certain lattice tilings. This criterion depends on a prime \(p\) and a tile \(B\). For \(p = 3\) and \(B\) being a Lee ball we recover the criterion of Zhang and Ge.Properties and parameters of codes from line graphs of circulant graphshttps://www.zbmath.org/1472.940852021-11-25T18:46:10.358925Z"Seneviratne, Pani"https://www.zbmath.org/authors/?q=ai:seneviratne.pani"Melendez, Jennifer D."https://www.zbmath.org/authors/?q=ai:melendez.jennifer-d"Westbrooks, Alexander N."https://www.zbmath.org/authors/?q=ai:westbrooks.alexander-nThe rows of an adjacency matrix of a graph may by viewed as codewords of a binary code, and the linear code generated by these may be studied. This is here done for line graphs of circulant graphs. Some good and even optimal (known) linear codes can be obtained in this way. The paper contains some old results on properties of codes of this type as well as a couple of proofs of new results.On transitive uniform partitions of \(F^n\) into binary Hamming codeshttps://www.zbmath.org/1472.940862021-11-25T18:46:10.358925Z"Solov'eva, Faina Ivanovna"https://www.zbmath.org/authors/?q=ai:soloveva.faina-ivanovnaSummary: We investigate transitive uniform partitions of the vector space \(F^n\) of dimension \(n\) over the Galois field \(\mathrm{GF}(2)\) into cosets of Hamming codes. A partition \(P^n=\{H_0,H_1 + e_1,\dots, H_n+ e_n\}\) of \(F^n\) into cosets of Hamming codes \(H_0,H_1,\dots, H_n\) of length \(n\) is said to be uniform if the intersection of any two codes \(H_i\) and \(H_j\), \(i,j\in\{0,1,\dots,n\}\) is constant, here \(e_i\) is a binary vector in \(F_n\) of weight 1 with one in the \(i\)th coordinate position. For any \(n= 2^m-1\), \(m > 4\) we found a class of nonequivalent 2-transitive uniform partitions of \(F^n\) into cosets of Hamming codes.On the self-dual codes with an automorphism of order 5https://www.zbmath.org/1472.940872021-11-25T18:46:10.358925Z"Yankov, Nikolay"https://www.zbmath.org/authors/?q=ai:yankov.nikolay-ivanov|yankov.nikolay"Anev, Damyan"https://www.zbmath.org/authors/?q=ai:anev.damyanAuthors' abstract: For lengths \(60\), \(62\), and \(64\), by applying the method for constructing self-dual codes having an automorphism of odd prime order, we classify all optimal singly even self-dual codes with an automorphism of order \(5\) with \(12\) cycles. For the binary self-dual \([62, 31, 12]\) codes we have found five new values of the parameter in the weight enumerator thus doubling the number of know values. For length 64 we have found codes with 14 new parameter values for both known weight enumerators. By shortening all binary self-dual [60, 30, 12] codes having an automorphism of order 5 we construct many new \([58, 29, 10]\) self-dual codes. We have found a new value of the parameter in the weight enumerator of these codes.Complete weight enumerators for several classes of two-weight and three-weight linear codeshttps://www.zbmath.org/1472.940882021-11-25T18:46:10.358925Z"Zhu, Canze"https://www.zbmath.org/authors/?q=ai:zhu.canze"Liao, Qunying"https://www.zbmath.org/authors/?q=ai:liao.qunyingAuthors' summary: In this paper, for an odd prime \(p\), by extending \textit{C. Li} et al.'s construction [Appl. Algebra Eng. Commun. Comput. 28, No. 1, 11--30 (2017; Zbl 1384.94126)], several classes of two-weight and three-weight linear codes over the finite field \(\mathbb{F}_2\) are constructed from a defining set, and then their complete weight enumerators are determined by using Weil sums. Furthermore, we show that some examples of these codes are optimal or almost optimal with respect to the Griesmer bound. Our results generalize the corresponding results in [\textit{G. Jian} et al., Finite Fields Appl. 57, 92--107 (2019; Zbl 1448.94264); \textit{C. Li} et al., Appl. Algebra Eng. Commun. Comput. 28, No. 1, 11--30 (2017; Zbl 1384.94126)].On the existence and construction of maximum distance profile convolutional codeshttps://www.zbmath.org/1472.940892021-11-25T18:46:10.358925Z"Muñoz Castañeda, Ángel Luis"https://www.zbmath.org/authors/?q=ai:munoz-castaneda.angel-luis"Plaza-Martín, Francisco J."https://www.zbmath.org/authors/?q=ai:plaza-martin.francisco-joseSummary: In this paper, we study the conditions for a convolutional code to be MDP in terms of the size of the base field \(\mathbb{F}_q\) as well as the openness of the MDP property in a given family of convolutional codes. Given \((n,k,\delta)\), our main result is an explicit bound depending on \((n,k,\delta)\) such that if \(q\) is greater than this bound, there exists a \((n,k,\delta)\) MDP convolutional code. A similar result is also offered for complete MDP convolutional codes. We show that these bounds are much lower than that those appeared so far in the literature. Finally, we show an explicit and simple construction procedure for MDP convolutional Goppa codes of dimension one.Skew constacyclic codes over the local Frobenius non-chain rings of order 16https://www.zbmath.org/1472.940902021-11-25T18:46:10.358925Z"Aydin, Nuh"https://www.zbmath.org/authors/?q=ai:aydin.nuh"Cengellenmis, Yasemin"https://www.zbmath.org/authors/?q=ai:cengellenmis.yasemin"Dertli, Abdullah"https://www.zbmath.org/authors/?q=ai:dertli.abdullah"Dougherty, Steven T."https://www.zbmath.org/authors/?q=ai:dougherty.steven-t"Saltürk, Esengül"https://www.zbmath.org/authors/?q=ai:salturk.esengulSummary: We introduce skew constacyclic codes over the local Frobenius non-chain rings of order 16 by defining non-trivial automorphisms on these rings. We study the Gray images of these codes, obtaining a number of binary and quaternary codes with good parameters as images of skew cyclic codes over some of these rings.Application of constacyclic codes over the semi local ring \(F_{p^m} + vF_{p^m}\)https://www.zbmath.org/1472.940912021-11-25T18:46:10.358925Z"Bag, Tushar"https://www.zbmath.org/authors/?q=ai:bag.tushar"Dertli, Abdullah"https://www.zbmath.org/authors/?q=ai:dertli.abdullah"Cengellenmis, Yasemin"https://www.zbmath.org/authors/?q=ai:cengellenmis.yasemin"Upadhyay, Ashish K."https://www.zbmath.org/authors/?q=ai:upadhyay.ashish-kumarSummary: In this paper, we study the quantum codes over \(F_{p^m}\), which are obtained from \(( \lambda_1 + \lambda_2)\)-constacyclic codes over the semi local ring \(F_{p^m}+ vF_{p^m}\), where \(v^2 = 1, p\) is odd prime. We decompose a \(( \lambda_1 + \lambda_2)\)-constacyclic code over \(F_{p^m} + vF_{p^m}\) into two constacyclic codes over \(F_{p^m}\) such as \(( \lambda_1 + \lambda_2)\)-constacyclic and \(( \lambda_1- \lambda_2)\)-constacyclic. We give the necessary and sufficient condition that the \(( \lambda_1 + v \lambda_2)\)-constacyclic codes over \(F_{p^m} + vF_{p^m}\) contain their duals. We give some examples of non binary quantum codes.Quantum codes from skew constacyclic codes over the ring \(\mathbb{F}_q [u, v] \slash \langle u^2 - 1, v^2 - 1, u v - v u \rangle \)https://www.zbmath.org/1472.940922021-11-25T18:46:10.358925Z"Bag, Tushar"https://www.zbmath.org/authors/?q=ai:bag.tushar"Dinh, Hai Q."https://www.zbmath.org/authors/?q=ai:dinh.hai-quang"Upadhyay, Ashish K."https://www.zbmath.org/authors/?q=ai:upadhyay.ashish-kumar"Bandi, Ramakrishna"https://www.zbmath.org/authors/?q=ai:bandi.ramakrishna"Yamaka, Woraphon"https://www.zbmath.org/authors/?q=ai:yamaka.woraphonSummary: In this paper, we study quantum error-correcting codes from skew constacyclic codes over the ring \(R = \frac{\mathbb{F}_q [u, v]}{\langle u^2 - 1, v^2 - 1, u v - v u \rangle} \), where \(q = p^m\) for any odd prime \(p\) and positive integer \(m\). We decompose skew constacyclic codes over the ring \(R\) as a direct sum of skew constacyclic codes over \(\mathbb{F}_q\). Self-dual skew constacyclic codes over the ring \(R\) are characterized. Necessary and sufficient conditions for skew negacyclic and skew constacyclic codes to be dual-containing are obtained. As an application, we construct new quantum error-correcting codes from skew constacyclic codes over \(\mathbb{F}_q\).A note on constacyclic and skew constacyclic codes over the ring \(\mathbb{Z}_p [u,v]/\langle u^2-u,v^2-v,uv-vu\rangle \)https://www.zbmath.org/1472.940932021-11-25T18:46:10.358925Z"Bag, Tushar"https://www.zbmath.org/authors/?q=ai:bag.tushar"Islam, Habibul"https://www.zbmath.org/authors/?q=ai:islam.habibul"Prakash, Om"https://www.zbmath.org/authors/?q=ai:prakash.om"Upadhyay, Ashish K."https://www.zbmath.org/authors/?q=ai:upadhyay.ashish-kumarSummary: For odd prime \(p\), this paper studies \((1+(p-2)u)\)-constacyclic codes over the ring \(R= \mathbb{Z}_p [u,v]/\langle u^2-u,v^2-v,uv-vu\rangle \). We show that the Gray images of \((1+(p-2)u)\)-constacyclic codes over \(R\) are cyclic and permutation equivalent to a quasi cyclic code over \(\mathbb{Z}_p \). We derive the generators for \((1+(p-2)u)\)-constacyclic and principally generated \((1+(p-2)u)\)-constacyclic codes over \(R\). Among others, we extend our results for skew \((1+(p-2)u)\)-constacyclic codes over \(R\) and exhibit the relation between skew \((1+(p-2)u)\)-constacyclic codes with the other linear codes. Finally, as an application of our study, we compute several non trivial linear codes by using the Gray images of \((1+(p-2)u)\)-constacyclic codes over this ring \(R\).Quantum codes from cyclic codes over the ring \(F_p [u] / \langle u^3 - u \rangle \)https://www.zbmath.org/1472.940942021-11-25T18:46:10.358925Z"Bag, Tushar"https://www.zbmath.org/authors/?q=ai:bag.tushar"Upadhyay, Ashish K."https://www.zbmath.org/authors/?q=ai:upadhyay.ashish-kumar"Ashraf, Mohammad"https://www.zbmath.org/authors/?q=ai:ashraf.mohammad"Mohammad, Ghulam"https://www.zbmath.org/authors/?q=ai:mohammad.ghulamAn explicit expression for Euclidean self-dual cyclic codes over \(\mathbb{F}_{2^m} + u \mathbb{F}_{2^m}\) of length \(2^s\)https://www.zbmath.org/1472.940952021-11-25T18:46:10.358925Z"Cao, Yuan"https://www.zbmath.org/authors/?q=ai:cao.yuan"Cao, Yonglin"https://www.zbmath.org/authors/?q=ai:cao.yonglin"Dinh, Hai Q."https://www.zbmath.org/authors/?q=ai:dinh.hai-quang"Wang, Guidong"https://www.zbmath.org/authors/?q=ai:wang.guidong"Sirisrisakulchai, Jirakom"https://www.zbmath.org/authors/?q=ai:sirisrisakulchai.jirakomSummary: Let \(\mathbb{F}_{2^m}\) be the finite field of \(2^m\) elements and \(s\) be any positive integer. The existing literature only gives an effective calculation method to represent all distinct Euclidean self-dual cyclic codes of length \(2^s\) over the finite chain ring \(\mathbb{F}_{2^m} + u \mathbb{F}_{2^m} ( u^2 = 0 )\), such as in [the first author et al., Discrete Math. 342, No. 7, 2077--2091 (2019; Zbl 1412.94241)]. As a development of this topic, we provide an explicit expression for each of these self-dual cyclic codes, using binomial coefficients. The Gray image of any self-dual cyclic code over \(\mathbb{F}_{2^m} + u \mathbb{F}_{2^m}\) of length \(2^s\) is a self-dual \(2\)-quasi-cyclic code over \(\mathbb{F}_{2^m}\) of length \(2^{s + 1}\). In particular, we give a generator matrix for each of these self-dual \(2\)-quasi-cyclic codes over \(\mathbb{F}_{2^m} \).On \(\mathbb{F}_2 RS\)-cyclic codes and their applications in constructing optimal codeshttps://www.zbmath.org/1472.940962021-11-25T18:46:10.358925Z"Dinh, Hai Q."https://www.zbmath.org/authors/?q=ai:dinh.hai-quang"Pathak, Sachin"https://www.zbmath.org/authors/?q=ai:pathak.sachin"Bag, Tushar"https://www.zbmath.org/authors/?q=ai:bag.tushar"Upadhyay, Ashish Kumar"https://www.zbmath.org/authors/?q=ai:upadhyay.ashish-kumar"Bandi, Ramakrishna"https://www.zbmath.org/authors/?q=ai:bandi.ramakrishna"Yamaka, Woraphon"https://www.zbmath.org/authors/?q=ai:yamaka.woraphonSummary: Let \(R = \mathbb{F}_2 + u \mathbb{F}_2\) \(( u^2 = 0 )\) and \(S = \mathbb{F}_2 + u \mathbb{F}_2 + u^2 \mathbb{F}_2\) \(( u^3 = 0 )\) be two finite commutative chain rings. This paper studies \(\mathbb{F}_2 R S\)-cyclic codes, which are described as \(S [ x ]\)-submodules of the \(S [ x ]\)-module \(\mathbb{F}_2 [ x ] \slash \langle x^r - 1 \rangle \times R [ x ] \slash \langle x^s - 1 \rangle \times S [ x ] \slash \langle x^t - 1 \rangle \). We study their generator polynomials and the minimal generating sets. We classify each case of the generating sets separately and determine the size of each such case. Free \(\mathbb{F}_2 R S\)-cyclic codes and separable codes are discussed, and the structural properties of dual codes of free \(\mathbb{F}_2 R S\)-cyclic codes are investigated. Moreover, we determine the relationship between the generator polynomials of free \(\mathbb{F}_2 R S\)-cyclic codes and their duals. As applications, we provide several examples of optimal and near-optimal binary codes which are obtained from the Gray images of \(\mathbb{F}_2 R S\)-cyclic codes.Quantum codes derived from one-generator quasi-cyclic codes with Hermitian inner producthttps://www.zbmath.org/1472.940972021-11-25T18:46:10.358925Z"Lv, Jingjie"https://www.zbmath.org/authors/?q=ai:lv.jingjie"Li, Ruihu"https://www.zbmath.org/authors/?q=ai:li.ruihu"Wang, Junli"https://www.zbmath.org/authors/?q=ai:wang.junliSummary: In this paper, we consider a family of one-generator quasi-cyclic codes and their applications in quantum codes construction. We give a sufficient condition for self-orthogonality with respect to Hermitian inner product. By virtue of the well-known MacWilliams equations, some binary and ternary stabilizer quantum codes with good parameters are constructed. Furthermore, we present a lower bound on the Hermitian dual distance of these involved codes. As the computational results, some good stabilizer quantum codes over small finite fields are obtained.Further results on optimal ternary cyclic codeshttps://www.zbmath.org/1472.940982021-11-25T18:46:10.358925Z"Zha, Zhengbang"https://www.zbmath.org/authors/?q=ai:zha.zhengbang"Hu, Lei"https://www.zbmath.org/authors/?q=ai:hu.lei"Liu, Yan"https://www.zbmath.org/authors/?q=ai:liu.yan.5"Cao, Xiwang"https://www.zbmath.org/authors/?q=ai:cao.xiwangSummary: Let \(\mathcal{C}_{(u,v)}\) denote the ternary cyclic code with two nonzeros \(\alpha^u\) and \(\alpha^v\), where \(\alpha\) is a generator of \(\mathbb{F}_{3^m}^\ast\) and \(0\leq u\), \(v\leq 3^m-2\). In this paper, we present a sufficient condition such that \(\mathcal{C}_{(u,v)}\) is an optimal ternary cyclic code. Based on this condition, we get several classes of optimal ternary cyclic codes by choosing the proper \(u\) and \(v\). Moreover, we show that \(\mathcal{C}_{(\frac{3^m+1}{2},\frac{3^m-1}{2}+v)}\) and \(\mathcal{C}_{(1,v)}\) have the same optimality.Asymptotically good homological error correcting codeshttps://www.zbmath.org/1472.940992021-11-25T18:46:10.358925Z"McCullough, Jason"https://www.zbmath.org/authors/?q=ai:mccullough.jason"Newman, Heather"https://www.zbmath.org/authors/?q=ai:newman.heather-aSummary: Let \(\Delta\) be an abstract simplicial complex. We study classical homological error correcting codes associated to \(\Delta \), which generalize the cycle codes of simple graphs. It is well-known that cycle codes of graphs do not yield asymptotically good families of codes. We show that asymptotically good families of codes do exist for homological codes associated to simplicial complexes of dimension at least \(2\). We also prove general bounds and formulas for (co-)cycle and (co-)boundary codes for arbitrary simplicial complexes over arbitrary fields.Construction of girth-8 \textit{(3,L)}-QC-LDPC codes of smallest CPM size using column multipliershttps://www.zbmath.org/1472.941002021-11-25T18:46:10.358925Z"Singh, Jasvinder"https://www.zbmath.org/authors/?q=ai:singh.jasvinder"Gupta, Manish"https://www.zbmath.org/authors/?q=ai:gupta.manish-kumar"Bhullar, Jaskarn Singh"https://www.zbmath.org/authors/?q=ai:bhullar.jaskarn-singhSummary: In this paper, a new method for the construction of the exponent matrix of quasi-cyclic low-density parity-check (QC-LDPC) codes is proposed. The entries of the exponent matrix are based on the column multipliers. To find the column multipliers, a parameter \({S}_{{\alpha}}\) is defined which gives the value of column multiplier of the \({{\alpha}}\) th column. The proposed method reduced the complexity related to the formation of the exponent matrix and results in \textit{(3,L)}-QC-LDPC codes with girth at least eight, for \({{L} > 3} \). Also, a lower bound on the size of the circulant permutation matrix (CPM) for a QC-LDPC code is derived, and the codes constructed by this method are optimal to the given bound. Further, most of the codes constructed using this method are of smaller CPM size. Specifically, for \({{L} > 25} \), our constructed QC-LDPC codes have the shortest CPM size compared to the existing ones in the literature.