Let $d, n \in \mathbb{Z}^+$ such that $1\leq d \leq n$. A $d$-code $\mathcal{C} \subset \mathbb{F}_q^{n \times n}$ is a subset of order $n$ square matrices with the property that for all pairs of distinct elements in $\mathcal{C}$, the rank of their difference is greater than or equal to $d$. A $d$-code with as many as possible elements is called a maximum $d$-code. The integer $d$ is also called the minimum distance of the code. When $d<n$, a classical example of such an object is the so-called generalized Gabidulin code. There exist several classes of maximum $d$-codes made up respectively of symmetric, alternating and hermitian matrices. In this article we focus on such examples. Precisely, we determine their automorphism groups and solve the equivalence issue for them. Finally, we exhibit a maximum symmetric $2$-code which is not equivalent to the one with same parameters known so far.
We consider the {\it infinite Johnson graph} $J_{\infty}$ whose vertex set consists of all subsets $X\subset {\mathbb N}$ satisfying $|X|=|{\mathbb N}\setminus X|=\infty$ and whose edges are pairs of such subsets $X,Y$ satisfying $|X\setminus Y|=|Y\setminus X|=1$. An automorphism of $J_{\infty}$ is said to be {\it regular} if it is induced by a permutation on $\mathbb{N}$ or it is the composition of the automorphism induced by a permutation on $\mathbb{N}$ and the automorphism $X\to {\mathbb N}\setminus X$. The graph $J_{\infty}$ admits non-regular automorphisms. Our first result states that t…
Let $k \geq 1$ be an integer and $n=3k-1$. Let $\mathbb{Z}_n$ denote the additive group of integers modulo $n$ and let $C$ be the subset of $\mathbb{Z}_n$ consisting of the elements congruent to 1 modulo 3. The Cayley graph $Cay(\mathbb{Z}_n; C)$ is known as the Andr$\acute{a}$sfai graph And($k$). In this note, we determine the automorphism group of this graph. We will show that $Aut(And(k))$ is isomorphic with the dihedral group $\mathbb{D}_{2n}$.
We study translation-invariant additive equations of the form $\sum_{i=1}^s λ_i \mathbf{P}(\mathbf{n}_i) = 0$ in variables $\mathbf{n}_i \in \mathbb{Z}^d$, where the $λ_i$ are nonzero integers summing to zero, and $\mathbf{P}$ is a system of homogeneous polynomials such that the above equation is invariant by translation. We investigate the solvability of this equation in subsets of density $(\log N)^{-c(\mathbf{P},\mathbfλ)}$ of a large box $[N]^d$, via the energy increment method. We obtain positive results in roughly the number of variables currently needed to derive a count of the solution…
Let $Γ=(V,E)$ be a point-transitive reflexive relation. Let $v\in V$ and put $r=|Γ(v)|.$ Also assume $Γ^j(v)\cap Γ^{-}(v)=\{v\}$. Then $$ |Γ^{j} (v)\setminus Γ^{j-1} (v)| \ge r-1.$$ In particular we have $ |Γ^{j} (v)| \ge 1+(r-1)j.$ The last result confirms a recent conjecture of Seymour in the case vertex-transitive graphs. Also it gives a short proof for the validity of the Caccetta-Häggkvist conjecture for vertex-transitive graphs and generalizes an additive result of Shepherdson.
We study the size of the automorphism group of two different types of random trees: Galton--Watson trees and rooted Pólya trees. In both cases, we prove that it asymptotically follows a log-normal distribution and provide asymptotic formulas for mean and variance of the logarithm of the size of the automorphism group. While the proof for Galton--Watson trees mainly relies on probabilistic arguments and a general result on additive tree functionals, generating functions are used in the case of rooted Pólya trees. We also show how to extend the results to some classes of unrooted trees.
A $Z_2Z_4$-linear Hadamard code of length $α+2β=2^t$ is a binary Hadamard code which is the Gray map image of a $Z_2Z_4$-additive code with $α$ binary coordinates and $β$ quaternary coordinates. It is known that there are exactly $[(t-1)/2]$ and $[t/2]$ nonequivalent $Z_2Z_4$-linear Hadamard codes of length $2^t$, with $α=0$ and $α\not=0$, respectively, for all $t\geq 3$. In this paper, it is shown that each $Z_2Z_4$-linear Hadamard code with $α=0$ is equivalent to a $Z_2Z_4$-linear Hadamard code with $α\not=0$; so there are only $[t/2]$ nonequivalent $Z_2Z_4$-linear Hadamard codes of length $…
We consider a family of 2-step nilpotent Lie algebras associated to uniform complete graphs on odd number of vertices. We prove that the symmetry group of such a graph is the holomorph of the additive cyclic group $\Z_n$. Moreover, we prove that the (Lie) automorphism group of the corresponding nilpotent Lie algebra contains the dihedral group of order $2n$ as a subgroup.
Let $Γ_{k}(V)$ be the Grassmann graph formed by $k$-dimensional subspaces of an $n$-dimensional vector space over the finite field ${\mathbb F}_{q}$ consisting of $q$ elements and $1<k<n-1$. Denote by $Γ(n,k)_q$ the restriction of the Grassmann graph to the set of all non-degenerate linear $[n,k]_q$ codes. We describe maximal cliques of the graph $Γ(n,k)_q$ and show that every automorphism of this graph is induced by a monomial semilinear automorphism of $V$.
This paper is devoted to the problem of classification of ${\rm AT4}(p,p+2,r)$-graphs. There is a unique ${\rm AT4}(p,p+2,r)$-graph with $p=2$, namely, the distance-transitive Soicher graph with intersection array $\{56, 45, 16, 1;1, 8, 45, 56\}$, whose local graphs are isomorphic to the Gewirtz graph. It is still unknown whether an ${\rm AT4}(p,p+2,r)$-graph with $p>2$ exists. The local graphs of each ${\rm AT4}(p,p+2,r)$-graph are strongly regular with parameters $((p+2)(p^2+4p+2),p(p+3),p-2,p)$. In the present paper, we find an upper bound for the prime spectrum of an automorphism group of …
One of the prominent areas of research in graph theory is the degree-diameter problem, in which we seek to determine how many vertices a graph may have when constrained to a given degree and diameter. Different variants of this problem are obtained by considering further restrictions on the graph, such as whether it is directed, undirected, mixed, vertex-transitive etc. The currently best known extremal constructions are the Gómez graphs, both when considered as directed and undirected. As the Gómez graphs are similar to the Faber-Moore-Chen graphs, we give a natural generalisation of their de…
We consider the symmetry group of a $Z_2Z_4$-linear code with parameters of a $1$-perfect, extended $1$-perfect, or Preparata-like code. We show that, provided the code length is greater than $16$, this group consists only of symmetries that preserve the $Z_2Z_4$ structure. We find the orders of the symmetry groups of the $Z_2Z_4$-linear (extended) $1$-perfect codes. Keywords: additive codes, $Z_2Z_4$-linear codes, $1$-perfect codes, Preparata-like codes, automorphism group, symmetry group.
Inspired by the work of Zhou "On equivalence of maximum additive symmetric rank-distance codes" (2020) based on the paper of Schmidt "Symmetric bilinear forms over finite fields with applications to coding theory" (2015), we investigate the equivalence issue of maximum $d$-codes of Hermitian matrices. More precisely, in the space $\mathrm{H}_n(q^2)$ of Hermitian matrices over $\mathbb{F}_{q^2}$ we have two possible equivalence: the classical one coming from the maps that preserve the rank in $\mathbb{F}_{q^2}^{n\times n}$, and the one that comes from restricting to those maps preserving both t…
Balog and Wooley have recently proved that any subset $A$ of either real numbers or of a prime finite field can be decomposed into two parts $U$ and $V$, one of small additive energy and the other of small multiplicative energy. In the case of arbitrary finite fields, we obtain an analogue that under some natural restrictions for a rational function $f$ both the additive energies of $U$ and $f(V)$ are small. Our method is based on bounds of character sums which leads to the restriction $\# A > q^{1/2}$ where $q$ is the field size. The bound is optimal, up to logarithmic factors, when $\# A \ge…
A finite group R is a CI-group if, whenever S and T are subsets of R with the Cayley graphs Cay(R,S) and Cay(R,T) isomorphic, there exists an automorphism x of R with S^x=T. The classification of CI-groups is an open problem in the theory of Cayley graphs and is closely related to the isomorphism problem for graphs. This paper is a contribution towards this classification, as we show that every dihedral group of order 6p, with p>3 prime, is a CI-group.
We consider additive codes over GF(4) that are self-dual with respect to the Hermitian trace inner product. Such codes have a well-known interpretation as quantum codes and correspond to isotropic systems. It has also been shown that these codes can be represented as graphs, and that two codes are equivalent if and only if the corresponding graphs are equivalent with respect to local complementation and graph isomorphism. We use these facts to classify all codes of length up to 12, where previously only all codes of length up to 9 were known. We also classify all extremal Type II codes of leng…
Additive codes over GF(9) that are self-dual with respect to the Hermitian trace inner product have a natural application in quantum information theory, where they correspond to ternary quantum error-correcting codes. However, these codes have so far received far less interest from coding theorists than self-dual additive codes over GF(4), which correspond to binary quantum codes. Self-dual additive codes over GF(9) have been classified up to length 8, and in this paper we extend the complete classification to codes of length 9 and 10. The classification is obtained by using a new algorithm th…
