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An algebraic approach to association schemes and coding theory
Abstract
The present work is a contribution to the algebraic theory of association schemes, where special emphasis is put on concepts arising from the theory of error correcting codes and of some combinatorial designs. The main idea is to characterize a subset in a given association scheme by its distribution with respect to the relations of the scheme. This yields some powerful methods for the study of subsets whose specific properties can be expressed in terms of their distribution. Various theorems are obtained in this way about generalized concepts of codes and t designs.
... While a classical approach to estimate the maximum cardinality of a code is to use Delsarte's LP method (which was first introduced in the context of the Hamming metric [14]), this has not yet been developed for sum-rank metric codes. The idea behind this method is to consider a code as a subset of points of an association scheme, and then formulate an optimization problem where the objective is to maximize the size of a code subject to linear constraints derived by leveraging the properties of the association scheme. ...
... This, combined with the duality of linear programming, provides one of the most powerful methods for bounding the size of codes arising from association schemes. Delsarte's LP approach has already been successfully applied to several metrics, e.g., for Hamming codes [14], rank-metric codes [15], bilinear alternating forms [16], permutation codes [17], and Lee codes [3,39], but also for newer metrics like subspace codes [36]. ...
... In [19], it was shown that, for k = 3, using the LP (5) it is possible to obtain tight bounds for every Hamming graph H(r, 2), and it was also shown that for these graphs the Ratio-type bound coincides with Delsarte's LP bound [14]; see also Section 5. ...
We derive a linear programming bound on the maximum cardinality of error-correcting codes in the sum-rank metric. Based on computational experiments on relatively small instances, we observe that the obtained bounds outperform all previously known bounds.
... The thesis of Philippe Delsarte [9] was a landmark in coding theory and combinatorial design theory. Employing the language of association schemes, Delsarte's approach was one of the first to cast combinatorial questions in a linear algebraic framework where tools from matrix theory, orthogonal polynomials, and optimisation could be applied. ...
... Employing the language of association schemes, Delsarte's approach was one of the first to cast combinatorial questions in a linear algebraic framework where tools from matrix theory, orthogonal polynomials, and optimisation could be applied. In particular, Delsarte [9] characterised orthogonal arrays and block designs as 01-vectors orthogonal to specific eigenspaces of the Hamming scheme and Johnson scheme, respectively. It is natural, then, to look at analogous substructures in the q-analogues of these families of association schemes and other classical families of Q-polynomial distance-regular graphs [6]. ...
... While the Hamming and Johnson schemes, etc., have only integer eigenvalues, most association schemes have irrational eigenvalues. The most obvious situation where restriction to a subfield of the splitting field is necessary is that in which one applies Delsarte's linear programming bound [9,Sec. 3.2] to problems in non-symmetric association schemes: here the splitting field is not contained in R so one moves to the symmetrisation to obtain a scheme with only real eigenvalues. ...
Delsarte theory, more specifically the study of codes and designs in association schemes, has proved invaluable in studying an increasing assortment of association schemes in recent years. Tools motivated by the study of error-correcting codes in the Hamming scheme and combinatorial $t$-designs in the Johnson scheme apply equally well in association schemes with irrational eigenvalues. We assume here that we have a commutative association scheme with irrational eigenvalues and wish to study its Delsarte $T$-designs. We explore when a $T$-design is also a $T'$-design where $T'\supseteq T$ is controlled by the orbits of a Galois group related to the splitting field of the association scheme. We then study Delsarte designs in the association schemes of finite groups, with a detailed exploration of the dicyclic groups.
... The power of association schemes in combinatorics stems from the observation that interesting combinatorial structures can often be characterised as subsets of association schemes for which certain entries in the inner or dual distribution are equal to zero. Delsarte [7] calls these objects cliques and designs, respectively. We shall see that interesting subsets of GL(n, q) indeed are cliques or designs in the conjugacy class association scheme of GL(n, q). ...
... The following result gives a characterisation of subsets of GL(n, q) that are transitive on α-flags as a design in the corresponding conjugacy class association scheme in the sense of Delsarte [7]. ...
... Another way to approach Theorem 5.3 involves the so-called clique-coclique bound [7, Theorem 3.9] and a condition on designs and antidesigns [28,Corollary 3.3] for the conjugacy class association scheme of GL(n, q). These results, proved in [7,28] for the case of symmetric association schemes, also hold in general. ...
It is known that the notion of a transitive subgroup of a permutation group G extends naturally to subsets of G . We consider subsets of the general linear group $${{\,\textrm{GL}\,}}(n,q)$$ GL ( n , q ) acting transitively on flag-like structures, which are common generalisations of t -dimensional subspaces of $$\mathbb {F}_q^n$$ F q n and bases of t -dimensional subspaces of $$\mathbb {F}_q^n$$ F q n . We give structural characterisations of transitive subsets of $${{\,\textrm{GL}\,}}(n,q)$$ GL ( n , q ) using the character theory of $${{\,\textrm{GL}\,}}(n,q)$$ GL ( n , q ) and interpret such subsets as designs in the conjugacy class association scheme of $${{\,\textrm{GL}\,}}(n,q)$$ GL ( n , q ) . In particular we generalise a theorem of Perin on subgroups of $${{\,\textrm{GL}\,}}(n,q)$$ GL ( n , q ) acting transitively on t -dimensional subspaces. We survey transitive subgroups of $${{\,\textrm{GL}\,}}(n,q)$$ GL ( n , q ) , showing that there is no subgroup of $${{\,\textrm{GL}\,}}(n,q)$$ GL ( n , q ) with $$1<t<n$$ 1 < t < n acting transitively on t -dimensional subspaces unless it contains $${{\,\textrm{SL}\,}}(n,q)$$ SL ( n , q ) or is one of two exceptional groups. On the other hand, for all fixed t , we show that there exist nontrivial subsets of $${{\,\textrm{GL}\,}}(n,q)$$ GL ( n , q ) that are transitive on linearly independent t -tuples of $$\mathbb {F}_q^n$$ F q n , which also shows the existence of nontrivial subsets of $${{\,\textrm{GL}\,}}(n,q)$$ GL ( n , q ) that are transitive on more general flag-like structures. We establish connections with orthogonal polynomials, namely the Al-Salam–Carlitz polynomials, and generalise a result by Rudvalis and Shinoda on the distribution of the number of fixed points of the elements in $${{\,\textrm{GL}\,}}(n,q)$$ GL ( n , q ) . Many of our results can be interpreted as q -analogs of corresponding results for the symmetric group.
... One potential drawback of Delsarte's approach is the necessity to explicitly compute the parameters of the underlying association scheme, a task that is not always easy. The parameters have been successfully computed for the graphs associated with various metrics, including the Hamming metric [7], the rank metric [8,9], permutation codes [11], and the Lee metric [4,19], but also for newer metrics such as the subspace distance [18]. ...
... In [9], Delsarte and Goethals considered precisely the problem of finding an upper bound on A q (n, 2d) using Delsarte's linear programming (LP) method; see Section 2.3 for a brief explanation and [7,10] for a detailed treatment. This resulted in the following bound relating the parameters of an alternating code. ...
... In this subsection we give a short overview of Delsarte's linear program. For a detailed treatment, we refer the interested reader to [7,10]. ...
In this note we apply a spectral method to the graph of alternating bilinear forms. In this way, we obtain upper bounds on the size of an alternating rank-metric code for given values of the minimum rank distance. We computationally compare our results with Delsarte's linear programming bound, observing that they give the same value. For small values of the minimum rank distance, we are able to establish the equivalence of the two methods. The problem remains open for larger values.
... There exist many multivariate generalizations of the polynomials of the Askey scheme (see [16,35,12,13,18,7,17]). Some of these polynomials appear already in the context of association schemes, in the expression of the eigenvalues of the adjacency matrices [8,24,26,4,29,21]. The goal of this paper consists in generalizing the notion of P -polynomial association schemes to a larger subclass of association schemes such that these multivariate polynomials appear naturally. ...
... Symmetrization of association scheme with two classes. In [8,26], the symmetrization of an association scheme has been defined and it has been shown that the expressions of the eigenvalues of the associated adjacency matrices are given by the multivariate Krawtchouk polynomials. We shall show that the symmetrization of an association scheme with two classes is a bivariate P -polynomial association scheme of type (1/2, 1/2). ...
... which is called symmetrization, defines an association scheme [8]. An explicit form for the matrix A ij is the following: ...
... equitable 2-partitions, perfect 2-colorings, and intriguing sets [1], [2], [12]. The concept of a completely regular code, suggested by Delsarte [7], is traditionally considered in classical association schemes. For a survey on completely regular codes in Hamming and Johnson graphs we refer to [3]. ...
... It is well-known that the strength is determined by the largest eigenvalue of the code in the following manner. A completely regular code has strength i if its maximum eigenvalue, different from the graph valency, has index i + 1 as eigenvalue of the Hamming graph [7]. The completely regular codes of strength 0 in the Hamming graphs were classified up to isomorphism by Meyerowitz in [11]. ...
... In this section we recall one of the most known properties of the completely regular codes in Hamming graphs which relate it to an orthogonal array of certain "strength" [7]. A similar fact could be formulated for a more general class of graphs such as the Cartesian product of complete graphs with different orders, which are not necessarily distance-regular. ...
We obtain a classification of the completely regular codes with covering radius 1 and the second eigenvalue in the Hamming graphs H(3, q) up to q and intersection array. Due to works of Meyerowitz, Mogilnykh and Valyuzenich, our result completes the classifications of completely regular codes with covering radius 1 and the second eigenvalue in the Hamming graphs H(n, q) for any n and completely regular codes with covering radius 1 in H(3, q).
... In addition to the well-studied linear codes, Delsarte et al. presented an important family of codes known as additive codes in their work on association schemes [14]. Generally, additive codes are subgroups of the underlying abelian group. ...
... (ω 2 , ω 2 , ω 2 , ω 2 , 0, ω 2 , ω 2 , 0, [23,33,4] [ 23,11,8] Optimal code 0, 0, 0, ω 2 , 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0) 6 (ω 2 , 0, 0, 0, 0, 0, ω 2 , ω 2 , 0, 0, 0, ω 2 , 0, 0, ω 2 , ω 2 , (ω 2 , ω 2 , ω, 0, ω 2 , 0, 0, 0, ω 2 , 0, 0, ω 2 , 0, 0, [43,56,4] [43, 14,14] Optimal code ω 2 , ω, 0, 0, ω 2 , 0, 0, ω 2 , 0, 0, 0, ω 2 , 0, ω 2 , ω 2 , ω, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ) 10 (ω 2 , 0, 0, 0, ω 2 , ω 2 , 0, ω 2 , 0, ω 2 , 0, ω 2 , ω 2 , ω 2 , ω 2 , [73, 117,4] [73, 45, 10] Optimal code ω 2 , 0, 0, ω 2 , 0, 0, 0, 0, ω 2 , ω 2 , ω 2 , 0, ω 2 , 0, ω 2 , 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0) an EAQEC code [[n, k 1 +k 2 −n+c, min{d 1 , d 2 }; c]] q , c = rank(H 1 H T 2 ) is the required number of maximally entangled states. ...
... (ω 2 , ω 2 , ω 2 , ω 2 , 0, ω 2 , ω 2 , 0, [23,33,4] [ 23,11,8] Optimal code 0, 0, 0, ω 2 , 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0) 6 (ω 2 , 0, 0, 0, 0, 0, ω 2 , ω 2 , 0, 0, 0, ω 2 , 0, 0, ω 2 , ω 2 , (ω 2 , ω 2 , ω, 0, ω 2 , 0, 0, 0, ω 2 , 0, 0, ω 2 , 0, 0, [43,56,4] [43, 14,14] Optimal code ω 2 , ω, 0, 0, ω 2 , 0, 0, ω 2 , 0, 0, 0, ω 2 , 0, ω 2 , ω 2 , ω, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ) 10 (ω 2 , 0, 0, 0, ω 2 , ω 2 , 0, ω 2 , 0, ω 2 , 0, ω 2 , ω 2 , ω 2 , ω 2 , [73, 117,4] [73, 45, 10] Optimal code ω 2 , 0, 0, ω 2 , 0, 0, 0, 0, ω 2 , ω 2 , ω 2 , 0, ω 2 , 0, ω 2 , 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0) an EAQEC code [[n, k 1 +k 2 −n+c, min{d 1 , d 2 }; c]] q , c = rank(H 1 H T 2 ) is the required number of maximally entangled states. ...
... A subset I of vertices of Γ, 0 < |I| < v, is said to be intriguing with parameters (h 1 , h 2 ) if there exist constants h 1 and h 2 such that every vertex of I is adjacent to precisely h 1 vertices of I and every vertex of V (Γ) \ I is adjacent to precisely h 2 vertices of I. This concept has been introduced by Delsarte [32] in the more general framework of association schemes and investigated in different contexts by several authors [1,2,3,15,17,20,36,57]. If I is intriguing with parameters (h 1 , h 2 ), then (h 1 − h 2 − k)j I + h 2 j is an eigenvector of the adjacency matrix A with the eigenvalue h 1 − h 2 . ...
... There are 45 negative intriguing sets of size 12 fixed by a group of order 1152, any two of them have 3 or 6 points in common. The corresponding SRG has parameters (45,32,22,24) that is the complement of the point graph of H(3, 4). (2) In NU (3,25), there is a negative intriguing set I of size 105, fixed by the group A 7 . ...
... Regarding m-ovoids of Q + (3, 3), there is a unique class of ovoids, being the conic sections and two distinct examples of 2-ovoids: one of which is a pair of disjoint conics admitting a subgroup of G of order 16 and there is one more stabilized by a subgroup of G of order 64. The related DSRGs have parameters (144, 36, 10, 6, 10), (144, 71, 39, 38, 32), (144, 72,40,32,40) and(288, 72,20,12,20). ...
In this paper we outline a technique for constructing directed strongly regular graphs by using strongly regular graphs having a "nice" family of intriguing sets. Further, we investigate such a construction method for rank three strongly regular graphs having at most $45$ vertices. Finally, several examples of intriguing sets of polar spaces are provided.
... Using this link, a complete characterization of the parameters of the multifold 1-perfect codes in -ary Hamming graphs was obtained for all being a power of a prime [11]. The definitions and background on additive and survey of completely regular codes are given in [5], [3]. It is well known, see Delsarte [5] that an additive code is completely regular with covering radius 1 if and only if its dual is a one-weight code. ...
... The definitions and background on additive and survey of completely regular codes are given in [5], [3]. It is well known, see Delsarte [5] that an additive code is completely regular with covering radius 1 if and only if its dual is a one-weight code. We give a direct proof of this fact in Section 2. As an illustration for the approach behind Theorem 1, we give a parity check matrix (3) of a completely regular code of length 5 over F 4 with intersection array {14; 2} arising from the (1, 2) 2,3 2 -multispread from Figure 1. ...
... • a (20, 4) 3, 5 3 -multispread, consisting of 10 subspaces of dimension 2 and 28 subspaces of dimension 3; ...
The additive one-weight codes over a finite field of non-prime order are equivalent to special subspace coverings of the points of projective space, which we call multispreads. The current paper is devoted to characterization of the parameters of multispreads, which is equivalent to the characterization of the parameters of additive one-weight codes. We characterize those parameters for the case of the prime-square order of the field and make a partial characterization for the prime-cube case and the case of the fourth degree of a prime (including complete characterization for orders 8, 27, and 16).
... , r}. In [17] it was shown that for k = 3 it is possible to obtain tight bounds for every Hamming graph H(2, r) using this LP approach with minor polynomials, and it is also shown that in for these graphs the Ratio-Type bound coincides with Delsarte's LP bound [14]. ...
... Remark 18. We should note that, in order to be applied, Delsarte's LP bound [14] requires a symmetric association scheme, which is a special partition of Mat(n, m, F q ) × Mat(n, m, F q ) into n + 1 binary relations R 0 , R 1 , . . . , R n ∈ R. As part of the definition, for any R i , R j , R k ∈ R the number of z ∈ Mat(n, m, F q ) such that (x, z) ∈ R i and (z, y) ∈ R j is the same for any x, y such that (x, y) ∈ R k . ...
... The value of ϑ k usually provides a good bound on α k , and it can be estimated using Semidefinite Programming (SDP) as follows [21]: For a graph G on n vertices, let A = (a ij ) range over all n × n symmetric matrices such that a ii = 1 for any i and a ij = 1 for any distinct i, j such that respective vertices of G are non-adjacent. Then ϑ = min A λ max (A), where λ max (A) is the largest eigenvalue of A. In fact, it was shown in [35] that, for graphs derived from symmetric association schemes, the Lovász theta number coincides with the bound obtained through Delsarte's LP method [14]. Tables 1 and 2 contain examples of sum-rank-metric graphs for which the Lovász theta number ϑ k , the Ratio-Type bound, and the bounds from Theorems 6 and 7 are calculated. ...
We consider the problem of deriving upper bounds on the parameters of sum-rank-metric codes, with focus on their dimension and block length. The sum-rank metric is a combination of the Hamming and the rank metric, and most of the available techniques to investigate it seem to be unable to fully capture its hybrid nature. In this paper, we introduce a new approach based on sum-rank-metric graphs, in which the vertices are tuples of matrices over a finite field, and where two such tuples are connected when their sum-rank distance is equal to one. We establish various structural properties of sum-rank-metric graphs and combine them with eigenvalue techniques to obtain bounds on the cardinality of sum-rank-metric codes. The bounds we derive improve on the best known bounds for several choices of the parameters. While our bounds are explicit only for small values of the minimum distance, they clearly indicate that spectral theory is able to capture the nature of the sum-rank-metric better than the currently available methods. They also allow us to establish new non-existence results for (possibly nonlinear) MSRD codes.
... A code C with covering radius ρ is called t-regular (0 ≤ t ≤ ρ) if for all i = 0, . . . , ρ, B x,i depends only on i and d(x, C), for all x such that d(x, C) ≤ t (see for instance [9,12]). A code C is completely regular if it is ρ-regular (see [9]). ...
... , ρ, B x,i depends only on i and d(x, C), for all x such that d(x, C) ≤ t (see for instance [9,12]). A code C is completely regular if it is ρ-regular (see [9]). We refer to [5], for a comprehensive survey on completely regular codes. ...
A subset $C$ of the vertex set of a graph $\Gamma$ is said to be $(\alpha,\beta)$-regular if $C$ induces an $\alpha$-regular subgraph and every vertex outside $C$ is adjacent to exactly $\beta$ vertices in $C$. In particular, if $C$ is an $(\alpha,\beta)$-regular set in some Cayley sum graph of a finite group $G$ with connection set $S$, then $C$ is called an $(\alpha,\beta)$-regular set of $G$ and a $(0,1)$-regular set is called a perfect code of $G$. By Sq$(G)$ and NSq$(G)$ we mean the set of all square elements and non-square elements of $G$. As one of the main results in this note, we show that a subgroup $H$ of a finite abelian group $G$ is an $(\alpha,\beta)$-regular set of $G$, for each $0\leq \alpha \leq |$NSq$(G)\cap H|$ and $0\leq \beta \leq \mathcal{L}(H)$, where $\mathcal{L}(H)=|H|$, if Sq$(G) \subseteq H$ and $\mathcal{L}(H)=|$NSq$(G)\cap H|$, otherwise. As a consequence of our result we give a very brief proof for the main results in \cite{mama, ma}. Also, we consider the dihedral group $G=D_{2n} $ and for each subgroup $H $ of $G$, by giving an appropriate connection set $S$, we determine each possibility for $(\alpha, \beta)$, where $H$ is an $(\alpha,\beta)$-regular set of $G$.
... neous space G/K is Gr(m, n). A classical result due to Delsarte [9,10] says that the zonal spherical functions for Gr(m, n) is expressed in terms of a q-Hahn polynomial, which is a 3 φ 2 basic hypergeometric polynomial and obtained by a degeneration limit of a q-Racah polynomial (see Sect. 3.1 for details). ...
... We have ω(x; 0; q) = ω x (e) = 1 by [24, VII.1, (1.4)]. An explicit formula for the value ω(x; i; q) is given by Delsarte [9,10]. We explain it below, following the presentation by Dunkl [13] with some modification. ...
We introduce a certain discrete probability distribution Pn,m,k,l;q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_{n,m,k,l;q}$$\end{document} having non-negative integer parameters n, m, k, l and quantum parameter q which arises from a zonal spherical function of the Grassmannian over the finite field Fq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {F}_q$$\end{document} with a distinguished spherical vector. Using representation theoretic arguments and hypergeometric summation technique, we derive the presentation of the probability mass function by a single q-Racah polynomial, and also the presentation of the cumulative distribution function in terms of a terminating 4ϕ3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${}_4 \phi _3$$\end{document}-hypergeometric series.
... Let be a primitive strongly regular graph with parameters (v, k, λ, μ) and spectrum k 1 , r f , s g , where k > r > s and the exponents mean the multiplicities. Recall that a coclique (an independent set) C in has cardinality not exceeding vs/(s − k), due to Delsarte [8,Sect. 3.3] and Hoffman [16]. ...
... (27,16,10,8) with f = 6 and g = 20. Then c = 27(−2)/(−2 − 16) = 3. ...
In 2022, the second author found a prolific construction of strongly regular graphs, which is based on joining a coclique and a divisible design graph with certain parameters. The construction produces strongly regular graphs with the same parameters as the complement of the symplectic graph Sp(2d,q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textsf{Sp}(2d,q)$$\end{document}. In this paper, we determine the parameters of strongly regular graphs which admit a decomposition into a divisible design graph and a coclique attaining the Hoffman bound. In particular, it is shown that when the least eigenvalue of such a strongly regular graph is a prime power, its parameters coincide with those of the complement of Sp(2d,q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textsf{Sp}(2d,q)$$\end{document}. Furthermore, a generalization of the construction is discussed.
... For this purpose, we shall use Lloyd's Theorem, which was originally proved by Lloyd [24] if q is a prime power. The general case was proved independently by Bassalygo [25], Delsarte [26], and Lenstra [27]. The statement of that theorem is as follows. ...
Perfect error-correcting codes allow for an optimal transmission of information while guaranteeing error correction. For this reason, proving their existence has been a classical problem in both pure mathematics and information theory. Indeed, the classification of the parameters of e-error correcting perfect codes over q-ary alphabets was a very active topic of research in the late 20th century. Consequently, all parameters of perfect e-error-correcting codes were found if e≥3, and it was conjectured that no perfect 2-error-correcting codes exist over any q-ary alphabet, where q>3. In the 1970s, this was proved for q a prime power, for q=2r3s and for only seven other values of q. Almost 50 years later, it is surprising to note that there have been no new results in this regard and the classification of 2-error-correcting codes over non-prime power alphabets remains an open problem. In this paper, we use techniques from the resolution of the generalised Ramanujan–Nagell equation and from modern computational number theory to show that perfect 2-error-correcting codes do not exist for 172 new values of q which are not prime powers, substantially increasing the values of q which are now classified. In addition, we prove that, for any fixed value of q, there can be at most finitely many perfect 2-error-correcting codes over an alphabet of size q.
... i=0 . Association schemes of Latin squares and their Bose-Mesner algebras are discussed in [1,7]. General references for association schemes include [1,2,3]. ...
We describe the Terwilliger algebras of the four-class Latin-square association schemes arising from Cayley tables of Bol loops. We give some necessary conditions involving Terwilliger algebras for a quasigroup to be a Bol loop.
... [5]'s work on finite groups provides foundational knowledge on group theory, which is essential for understanding the algebraic structures underlying the Weisfeiler-Leman algorithm and its applications. [6] and [7] presents an algebraic framework for studying association schemes in coding theory, offering insights into their structural properties and connections to various mathematical concepts. [8]'s work on distance-regular graphs provides background knowledge on graph theory, which is relevant to understanding the properties of graphs and their relationships to algebraic structures. ...
In this paper, we explore the algebraic interpretation of the partitioning obtained by the 𝑚-dimensional Weisfeiler–Leman algorithm on the direct power Gm of a finite group G. We define and study a Schur ring over Gm, which provides insights into the structure of the group G. Our analysis reveals that this ring determines the group G up to isomorphism when m≥3. Furthermore, we demonstrate that as m increases, the Schur ring associated with the group of automorphisms of G acting on Gm emerges naturally. Surprisingly, we establish that finding the limit ring is polynomial-time equivalent to solving the group isomorphism problem. This paper presents a novel algebraic framework for understanding the behavior of the Weisfeiler–Leman algorithm and its implications for group theory and computational complexity.
... In essence, additive codes can employ different types of dualities, not limited to the standard ones, to achieve their specific coding objectives. Delsarte [10] introduced additive codes in 1973 using association schemes. For symmetric and non-symmetric dualities, an additive code satisfies the cardinality condition. ...
... In essence, additive codes can employ different types of dualities, not limited to the standard ones, to achieve their specific coding objectives. Delsarte [10] introduced additive codes in 1973 using association schemes. For symmetric and non-symmetric dualities, an additive code satisfies the cardinality condition. ...
... Then, Numerical examinations of the tightness of bounds on graph invariants, as given in Corollary 5.8, are provided in Example 5.21 for several strongly regular graphs. The upper bound on the independence number, presented in (5.30), is a formulation of the Delsarte and Hoffman bound, which was originally developed for regular graphs (see Section 3.3 in [152] and the survey in [153]). This formulation has been specialized in (5.30) to apply to strongly regular graphs. ...
This paper delves into three research directions, leveraging the Lovász ϑ-function of a graph. First, it focuses on the Shannon capacity of graphs, providing new results that determine the capacity for two infinite subclasses of strongly regular graphs, and extending prior results. The second part explores cospectral and nonisomorphic graphs, drawing on a work by Berman and Hamud (2024), and it derives related properties of two types of joins of graphs. For every even integer such that n≥14, it is constructively proven that there exist connected, irregular, cospectral, and nonisomorphic graphs on n vertices, being jointly cospectral with respect to their adjacency, Laplacian, signless Laplacian, and normalized Laplacian matrices, while also sharing identical independence, clique, and chromatic numbers, but being distinguished by their Lovász ϑ-functions. The third part focuses on establishing bounds on graph invariants, particularly emphasizing strongly regular graphs and triangle-free graphs, and compares the tightness of these bounds to existing ones. The paper derives spectral upper and lower bounds on the vector and strict vector chromatic numbers of regular graphs, providing sufficient conditions for the attainability of these bounds. Exact closed-form expressions for the vector and strict vector chromatic numbers are derived for all strongly regular graphs and for all graphs that are vertex- and edge-transitive, demonstrating that these two types of chromatic numbers coincide for every such graph. This work resolves a query regarding the variant of the ϑ-function by Schrijver and the identical function by McEliece et al. It shows, by a counterexample, that the ϑ-function variant by Schrijver does not possess the property of the Lovász ϑ-function of forming an upper bound on the Shannon capacity of a graph.
... Delsarte proved [7] that the clique number of a strongly regular graph G is at most 1− k s . A clique in a strongly regular graph whose size attains this bound is called a Delsarte clique. ...
A divisible design graph is a graph whose adjacency matrix is an incidence matrix of a (group) divisible design. Divisible design graphs were introduced in 2011 as a generalization of (v, k, λ)-graphs. Here we describe four new infinite families that can be obtained from the symplectic strongly regular graph Sp(2e, q) (q odd, e ≥ 2) by modifying the set of edges. To achieve this we need two kinds of spreads in P G(2e − 1, q) with respect to the associated symplectic form: the symplectic spread consisting of totally isotropic subspaces and, when e = 2, a special spread consisting of lines which are not totally isotropic. Existence of symplectic spreads is known, but the construction of a special spread for every odd prime power q is a major result of this paper. We have included relevant back ground from finite geometry, and when q = 3, 5 and 7 we worked out all possible special spreads.
... We have A = kJ + rE r + sE s , I = J n + E r + E s , and so { J n , E r , E s } is an orthogonal basis of idempotents of C[A X ]. We now mention a bound by Delsarte (see equation (3.25) of [11]) on cliques in strongly regular graphs. We state the formulation of this result as given in [19, Corollary 3.7.2]. ...
... These codes are a generalized version of linear codes, i.e., all linear codes are additive codes, but converse is not true; see [2]. For the first time in 1973, Delsarte et al. defined additive codes in terms of association schemes, and with the help of additive codes, they gave constructions of schemes with good performance (see, for instance, [6,7]). In [5], Calderbank et al. used the trace inner product to construct a quantum code from a given additive code over F 4 . ...
Additive codes have gained importance in algebraic coding theory due to their applications in quantum error correction and quantum computing. The article begins by developing some properties of Additive Complementary Dual (ACD) codes with respect to arbitrary dualities over finite abelian groups. Further, we introduce a subclass of non-symmetric dualities referred to as the skew-symmetric dualities. Then, we precisely count symmetric and skew-symmetric dualities over finite fields. Two conditions have been obtained: one is a necessary and sufficient condition, and the other is a necessary condition. The necessary and sufficient condition is for an additive code to be an ACD code over arbitrary dualities. The necessary condition is on a generator matrix of an ACD code over skew-symmetric dualities. We provide bounds for the highest possible minimum distance of ACD codes over skew-symmetric dualities. Finally, we find some new quaternary ACD codes over non-symmetric dualities with better parameters than the symmetric ones.
... In 1973, additive codes were first defined by Delsarte [13,14] in terms of association schemes. Generally, an additive code is defined as a subgroup of the underlying abelian group. ...
... Let us recall the notion of association schemes briefly. See Bannai-Ito [1] and Delsarte [4] for details. We summarize basic terminologies. ...
A wreath product is a method to construct an association scheme from two association schemes. We determine the automorphism group of a wreath product. We show a known result that a wreath product is Schurian if and only if both components are Schurian, which yields large families of non-Schurian association schemes and non-Schurian $S$-rings. We also study iterated wreath products. Kernel schemes by Martin and Stinson are shown to be iterated wreath products of class-one association schemes. The iterated wreath products give examples of projective systems of non-Schurian association schemes, with an explicit description of primitive idempotents.
... We note here the influential thesis of Delsarte (1973), which contains an investigation of subsets in arbitrary association schemes whose characteristic vectors are orthogonal to some strata, or common eigenspaces, of the scheme. In particular, Delsarte defined the notion of Q-polynomial schemes, in which the strata have a natural order 0 , 1 , 2 , . . . . ...
... There are many Hoffman-type ratio bounds, which are useful in graph theory, coding theory and extremal combinatorics [8,9,10,12,13]. Some generalized Hoffman ratio bounds involving adjacency eigenvalues, Laplacian eigenvalues and normalized Laplacian eigenvalues of graphs can be found in [14,17,18,20]. ...
The Hoffman ratio bound, Lovász theta function, and Schrijver theta function are classical upper bounds for the independence number of graphs, which are useful in graph theory, extremal combinatorics, and information theory. By using generalized inverses and eigenvalues of graph matrices, we give bounds for independence sets and the independence number of graphs. Our bounds unify the Lovász theta function, Schrijver theta function, and Hoffman-type bounds, and we obtain the necessary and sufficient conditions of graphs attaining these bounds. Our work leads to some simple structural and spectral conditions for determining a maximum independent set, the independence number, the Shannon capacity, and the Lovász theta function of a graph.
... Proposition 6 (Theorem 5.13 in [6]). Let C be a distance-d code in a distance-regular graph, and let R be the number of non-zero dual distances of C (the external distance of C). ...
Классифицированы все линейные полностью регулярные коды с радиусом покрытия ρ = 2, дуальные коды которых являются антиподальными. Для этого вначале приводится ряд свойств для таких дуальных кодов, являющихся кодами с двумя расстояниями d и n.
Let ${\mathcal X} = (X, \{R_i\}_{i=0}^d)$ denote a symmetric association scheme. Fix an ordering $\{E_i\}_{i=0}^d$ of the primitive idempotents of $\mathcal{X}$, and let $P$ (resp.\ $Q$) denote the corresponding first eigenmatrix (resp.\ second eigenmatrix) of $\mathcal X$. The scheme $\mathcal X$ is said to be formally self-dual (with respect to the ordering $\{E_i\}_{i=0}^d$) whenever $P=Q$. We define $\mathcal X$ to be numerically self-dual (with respect to the ordering $\{E_i\}_{i=0}^d$) whenever the intersection numbers and Krein parameters satisfy $p^h_{i,j} =q^h_{i,j}$ for $0 \leq h,i,j \leq d$. It is known that with respect to the ordering $\{E_i\}_{i=0}^d$, formal self-duality implies numerical self-duality. This raises the following question: is it possible that with respect to the ordering $\{E_i\}_{i=0}^d$, $\mathcal X$ is numerically self-dual but not formally self-dual? This is possible as we will show. We display an example of a symmetric association scheme and an ordering the primitive idempotents with respect to which the scheme is numerically self-dual but not formally self-dual. We have the following additional results about self-duality. Assume that $\mathcal X$ is $P$-polynomial. We show that the following are equivalent: (i) $\mathcal X$ is formally self-dual with respect to the ordering $\{E_i\}_{i=0}^d$; (ii) $\mathcal X$ is numerically self-dual with respect to the ordering $\{E_i\}_{i=0}^d$. Assume that the ordering $\{E_i\}_{i=0}^d$ is $Q$-polynomial. We show that the following are equivalent: (i) $\mathcal X$ is formally self-dual with respect to the ordering $\{E_i\}_{i=0}^d$; (ii) $\mathcal X$ is numerically self-dual with respect to the ordering $\{E_i\}_{i=0}^d$.
Strongly regular graphs (SRGs) provide a fertile area of exploration in algebraic combinatorics, integrating techniques in graph theory, linear algebra, group theory, finite fields, finite geometry, and number theory. Of particular interest are those SRGs with a large automorphism group. If an automorphism group acts regularly (sharply transitively) on the vertices of the graph, then we may identify the graph with a subset of the group, a partial difference set (PDS), which allows us to apply techniques from group theory to examine the graph. Much of the work over the past four decades has concentrated on abelian PDSs using the powerful techniques of character theory. However, little work has been done on nonabelian PDSs. In this paper we point out the existence of genuinely nonabelian PDSs, that is, PDSs for parameter sets where a nonabelian group is the only possible regular automorphism group. We include methods for demonstrating that abelian PDSs are not possible for a particular set of parameters or for a particular SRG. Four infinite families of genuinely nonabelian PDSs are described, two of which—one arising from triangular graphs and one arising from Krein covers of complete graphs constructed by Godsil—are new. We also include a new nonabelian PDS found by computer search and present some possible future directions of research.
This is an introduction to representation theory and harmonic analysis on finite groups. This includes, in particular, Gelfand pairs (with applications to diffusion processes à la Diaconis) and induced representations (focusing on the little group method of Mackey and Wigner). We also discuss Laplace operators and spectral theory of finite regular graphs. In the last part, we present the representation theory of GL(2, Fq), the general linear group of invertible 2 × 2 matrices with coefficients in a finite field with q elements. More precisely, we revisit the classical Gelfand–Graev representation of GL(2, Fq) in terms of the so-called multiplicity-free triples and their associated Hecke algebras. The presentation is not fully self-contained: most of the basic and elementary facts are proved in detail, some others are left as exercises, while, for more advanced results with no proof, precise references are provided.
A set of k orthonormal bases of C d is called mutually unbiased if | ⟨ e , f ⟩ | 2 = 1 / d whenever e and f are basis vectors in distinct bases. A natural question is for which pairs ( d , k ) there exist k mutually unbiased bases in dimension d . The (well-known) upper bound k ≤ d + 1 is attained when d is a power of a prime. For all other dimensions it is an open problem whether the bound can be attained. Navascués, Pironio, and Acín showed how to reformulate the existence question in terms of the existence of a certain C ∗ -algebra. This naturally leads to a noncommutative polynomial optimization problem and an associated hierarchy of semidefinite programs. The problem has a symmetry coming from the wreath product of S d and S k .We exploit this symmetry (analytically) to reduce the size of the semidefinite programs making them (numerically) tractable. A key step is a novel explicit decomposition of the S d ≀ S k -module C ( [ d ] × [ k ] ) t into irreducible modules. We present numerical results for small d , k and low levels of the hierarchy. In particular, we obtain sum-of-squares proofs for the (well-known) fact that there do not exist d + 2 mutually unbiased bases in dimensions d = 2 , 3 , 4 , 5 , 6 , 7 , 8 . Moreover, our numerical results indicate that a sum-of-squares refutation, in the above-mentioned framework, of the existence of more than 3 MUBs in dimension 6 requires polynomials of total degree at least 12 .
Получены нижние и верхние оценки минимального числа ребер в индуцированных подграфах с $l$ вершинами графа $G(n,3,1)$, где $l \sim cn^2$. Полученные результаты улучшают ранее доказанные оценки этой величины в данном режиме. Библиография: 16 названий.
We propose concatenated and switching methods for the construction of single-error-correcting perfect and diameter codes in the Lee metric. We analyze ranks and kernels of diameter perfect codes obtained by the switching construction.
We classify all linear completely regular codes which have covering radius \(\rho=2\) and whose dual are antipodal. For this, we firstly show several properties of such dual codes, which are two-weight codes with weights \(d\) and \(n\).
Let $\mathbb{F}$ denote a field, and let $V$ denote a vector space over $\mathbb{F}$ with finite positive dimension. A Leonard pair on $V$ is an ordered pair of diagonalizable $\mathbb{F}$-linear maps $A: V \to V$ and $A^* : V \to V$ that each act on an eigenbasis for the other in an irreducible tridiagonal fashion. Let $A,A^*$ denote a Leonard pair on $V$. Let $\{v_i\}_{i=0}^d$ denote an eigenbasis for $A^*$ on which $A$ acts in an irreducible tridiagonal fashion. For $0 \leq i \leq d$, define an $\mathbb{F}$-linear map $E^*_i : V \to V$ such that $E^*_i v_i = v_i$ and $E^*_i v_j = 0$ if $j \neq i$ $(0 \leq j \leq d)$. The map $F = \sum_{i=0}^d E^*_i A E^*_i$ is called the flat part of $A$. The Leonard pair $A,A^*$ is bipartite whenever $F=0$. The Leonard pair $A,A^*$ is said to be near-bipartite whenever the pair $A-F, A^*$ is a Leonard pair on $V$. In this case, the Leonard pair $A-F, A^*$ is bipartite and called the bipartite contraction of $A,A^*$. Let $B,B^*$ denote a bipartite Leonard pair on $V$. By a near-bipartite expansion of $B,B^*$, we mean a near-bipartite Leonard pair on $V$ with bipartite contraction $B,B^*$. In the present paper, we have three goals. Assuming $\mathbb{F}$ is algebraically closed, (i) we classify up to isomorphism the near-bipartite Leonard pairs over $\mathbb{F}$; (ii) for each near-bipartite Leonard pair over $\mathbb{F}$ we describe its bipartite contraction; (iii) for each bipartite Leonard pair over $\mathbb{F}$ we describe its near-bipartite expansions. Our classification (i) is summarized as follows. We identify two families of Leonard pairs, said to have Krawtchouk type and dual $q$-Krawtchouk type. A Leonard pair of dual $q$-Krawtchouk type is said to be reinforced whenever $q^{2i} \neq -1$ for $1 \leq i \leq d-1$. A Leonard pair $A,A^*$ is said to be essentially bipartite whenever the flat part of $A$ is a scalar multiple of the identity. Assuming $\mathbb{F}$ is algebraically closed, we show that a Leonard pair $A,A^*$ over $\mathbb{F}$ with $d \geq 3$ is near-bipartite if and only if at least one of the following holds: (i) $A,A^*$ is essentially bipartite; (ii) $A,A^*$ has reinforced dual $q$-Krawtchouk type; and (iii) $A,A^*$ has Krawtchouk type.
Maximum rank‐distance (MRD) codes are (not necessarily linear) maximum codes in the rank‐distance metric space on ‐by‐ matrices over a finite field . They are diameter perfect and have the cardinality if . We define switching in MRD codes as the replacement of special MRD subcodes by other subcodes with the same parameters. We consider constructions of MRD codes admitting switching, such as punctured twisted Gabidulin codes and direct‐product codes. Using switching, we construct a huge class of MRD codes whose cardinality grows doubly exponentially in if the other parameters (, the code distance) are fixed. Moreover, we construct MRD codes with different affine ranks and aperiodic MRD codes.
A finite set of the Euclidean space is called an s-distance set provided that the number of Euclidean distances in the set is s. Determining the largest possible s-distance set for the Euclidean space of a given dimension is challenging. This problem was solved only when dealing with small values of s and dimensions. Lisoněk (J Combin Theory Ser A 77(2):318–338, 1997) achieved the classification of the largest 2-distance sets for dimensions up to 7, using computer assistance and graph representation theory. In this study, we consider a theory analogous to these results of Lisoněk for the pseudo-Euclidean space \(\mathbb {R}^{p,q}\). We consider an s-indefinite-distance set in a pseudo-Euclidean space that uses the value
instead of the Euclidean distance. We develop a representation theory for symmetric matrices in the context of s-indefinite-distance sets, which includes or improves the results of Euclidean s-distance sets with large s values. Moreover, we classify the largest possible 2-indefinite-distance sets for small dimensions.
Given a graph , we let denote the sum of the squares of the positive eigenvalues of the adjacency matrix of , and we similarly define . We prove that and thus strengthen a result of Ando and Lin, who showed the same lower bound for the chromatic number . We in fact show a stronger result wherein we give a bound using the eigenvalues of and whenever has a homomorphism to an edge‐transitive graph . Our proof utilizes ideas motivated by association schemes.
We propose concatenation and switching methods for the construction of single-error-correcting perfect and diameter codes in the Lee metric. We analyze ranks and kernels of diameter perfect codes obtained by the switching construction.
Let \(\text {S}\) be a finite set, each element of which receives a color. A rainbow t-set of \(\text {S}\) is a t-subset of \(\text {S}\) in which different elements receive different colors. Let \(\left( {\begin{array}{c}\text {S}\\ t\end{array}}\right) \) denote the set of all rainbow t-sets of \(\text {S}\), let \(\left( {\begin{array}{c}\text {S}\\ \le t\end{array}}\right) \) represent the union of \(\left( {\begin{array}{c}\text {S}\\ i\end{array}}\right) \) for \(i=0,\ldots , t\), and let \(2^\text {S}\) stand for the set of all rainbow subsets of \(\text {S}\). The rainbow inclusion matrix \(\mathcal {W}^{\text {S}}\) is the \(2^\text {S}\times 2^{\text {S}}\) (0, 1) matrix whose (T, K)-entry is one if and only if \(T\subseteq K\). We write \(\mathcal {W}_{t,k}^{\text {S}}\) and \(\mathcal {W}_{\le t,k}^{\text {S}}\) for the \(\left( {\begin{array}{c}\text {S}\\ t\end{array}}\right) \times \left( {\begin{array}{c}\text {S}\\ k\end{array}}\right) \) submatrix and the \(\left( {\begin{array}{c}\text {S}\\ \le t\end{array}}\right) \times \left( {\begin{array}{c}\text {S}\\ k\end{array}}\right) \) submatrix of \(\mathcal {W}^{\text {S}}\), respectively, and so on. We determine the diagonal forms and the ranks of \(\mathcal {W}_{t,k}^{\text {S}}\) and \(\mathcal {W}_{\le t,k}^{\text {S}}\). We further calculate the singular values of \(\mathcal {W}_{t,k}^{\text {S}}\) and construct accordingly a complete system of \((0,\pm 1)\) eigenvectors for them when the numbers of elements receiving any two given colors are the same. Let \(\mathcal {D}^{\text {S}}_{t,k}\) denote the integral lattice orthogonal to the rows of \(\mathcal {W}_{\le t,k}^{\text {S}}\) and let \(\overline{\mathcal {D}}^{\text {S}}_{t,k}\) denote the orthogonal lattice of \(\mathcal {D}^{\text {S}}_{t,k}\). We make use of Frankl rank to present a \((0,\pm 1)\) basis of \(\mathcal {D}^{\text {S}}_{t,k}\) and a (0, 1) basis of \(\overline{\mathcal {D}}^{\text {S}}_{t,k}\). For any commutative ring R, those nonzero functions \(f\in R^{2^{\text {S}}}\) satisfying \(\mathcal {W}_{t,\ge 0}^{\text {S}}f=0\) are called null t-designs over R, while those satisfying \(\mathcal {W}_{\le t,\ge 0}^{\text {S}}f=0\) are called null \((\le t)\)-designs over R. We report some observations on the distributions of the support sizes of null designs as well as the structure of null designs with extremal support sizes.
Simplex and MacDonald codes have received significant attention from researchers since the inception of coding theory. In this work, we present the construction of linear torsion codes for simplex and MacDonald codes over the ring \({\mathcal {R}}={\mathcal {R}}_{1}{\mathcal {R}}_{2}{\mathcal {R}}_{3}\). We have introduced a novel family of linear codes over \({\mathbb {F}}_{p}\). These codes have been extensively examined with respect to their properties, such as code minimality, weight distribution, and their applications in secret sharing schemes. In addition to this investigation, we have discovered that these codes are also applicable to the association schemes of linear torsion codes for simplex and MacDonald codes over. \({\mathcal {R}}={\mathcal {R}}_{1}{\mathcal {R}}_{2}{\mathcal {R}}_{3}\).
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