Let $\mathcal{S}_n$ denote the set of permutations of $\{1,2,\dots,n\}$. The function $f(n,s)$ is defined to be the minimum size of a subset $S\subseteq \mathcal{S}_n$ with the property that for any $ρ\in \mathcal{S}_n$ there exists some $σ\in S$ such that the Hamming distance between $ρ$ and $σ$ is at most $n-s$. The value of $f(n,2)$ is the subject of a conjecture by Kézdy and Snevily, which implies several famous conjectures about latin squares. We prove that the odd $n$ case of the Kézdy-Snevily Conjecture implies the whole conjecture. We also show that $f(n,2)>3n/4$ for all $n$, that $s!< f(n,s)< 3s!(n-s)\log n$ for $1\leq s\leq n-2$ and that \[f(n,s)>\left\lfloor \frac{2+\sqrt{2s-2}}{2}\right\rfloor \frac{n}{2}\] if $s\geq 3$.
A family $\mathcal{F}$ of subsets of a set $X$ is $t$-intersecting if $\vert A_i \cap A_j \vert \geq t$ for every $A_i, \; A_j \in \mathcal{F}$. We study intersecting families in the Hamming geometry. Given $X=\mathbb{F}_q^3$ a vector space over the finite field $\mathbb{F}_q$, consider a family where each $A_i$ is an extended ball, that is, $A_i$ is the union of all balls centered in the scalar multiples of a vector. The geometric behavior of extended balls is discussed. As the main result, we investigate a ``large" arrangement of vectors whose extended balls are ``highly intersecting". Consi…
The main result of this work is the characterization of the covering relations of the Bruhat order of the maximal parabolic quotients of type B. Our approach is mainly combinatorial and is based in the pattern of the corresponding permutations also called signed $k$-Grassmannians permutations. We obtain that a covering relation can be classified in four different pairs of permutations. This answers a question raised by Ikeda and Matsumura providing a nice combinatorial model for maximal parabolic quotients of type B.
We recover the first linear programming bound of McEliece, Rodemich, Rumsey, and Welch for binary error-correcting codes and designs via a covering argument. It is possible to show, interpreting the following notions appropriately, that if a code has a large distance, then its dual has a small covering radius and, therefore, is large. This implies the original code to be small. We also point out (in conjunction with further work) that this bound is a natural isoperimetric constant of the Hamming cube, related to its Faber-Krahn minima. While our approach belongs to the general framework of Del…
The covering radius of permutation group codes are studied in this paper with $l_{\infty}$-metric. We determine the covering radius of the $(p,q)$-type group, which is a direct product of two cyclic transitive groups. We also deduce the maximum covering radius among all the relabelings of this group under conjugation, that is, permutation groups with the same algebraic structure but with relabelled members. Finally, we give a lower bound of the covering radius of the dihedral group code, which differs from the trivial upper bound by a constant at most one. This improves the result of Karni and…
Consider the metric space $\mathcal{C}$ consisting of the $n$-dimensional Boolean cube equipped with the Hamming distance. A weak isometry of $\mathcal{C}$ is a permutation of $\mathcal{C}$ preserving a given subset of Hamming distances. In \cite{Krasin} Krasin showed that in most cases preserving a single Hamming distance forces a weak isometry to be an isometry. In this article we study those weak isometries that are not automatically an isometry, providing a complete classification of weak isometries of $\mathcal{C}$.
Let S_n be the set of all permutations on [n]:={1,2,....,n}. We denote by kappa_n the smallest cardinality of a subset A of S_{n+1} that "covers" S_n, in the sense that each pi in S_n may be found as an order-isomorphic subsequence of some pi' in A. What are general upper bounds on kappa_n? If we randomly select nu_n elements of S_{n+1}, when does the probability that they cover S_n transition from 0 to 1? Can we provide a fine-magnification analysis that provides the "probability of coverage" when nu_n is around the level given by the phase transition? In this paper we answer these questions …
Let M(n, d) be the maximum size of a permutation array on n symbols with pairwise Hamming distance at least d. We use various combinatorial, algebraic, and computational methods to improve lower bounds for M(n, d). We compute the Hamming distances of affine semilinear groups and projective semilinear groups, and unions of cosets of AGL(1,q) and PGL(2,q) with Frobenius maps to obtain new, improved lower bounds for M(n,d). We give new randomized algorithms. We give better lower bounds for M(n,d) also using new theorems concerning the contraction operation. For example, we prove a quadratic lower…
The covering radius of a subset $C$ of the symmetric group $\mathrm{S}_n$ is the maximal Hamming distance of an element of $\mathrm{S}_n$ from $C$. This note determines the covering radii of the finite projective general linear groups. It turns out that the covering radius of $\mathrm{PGL}_2(q)$ is $q-2$ if $q$ is even, and is $q-3$ if $q$ is odd.
A binary code with covering radius $R$ is a subset $C$ of the hypercube $Q_n=\{0,1\}^n$ such that every $x\in Q_n$ is within Hamming distance $R$ of some codeword $c\in C$, where $R$ is as small as possible. For a fixed coordinate $i\in[n]$, define $C(b,i)$, for $b=0,1$, to be the set of codewords with a $b$ in the $i$th position. Then $C$ is normal if there exists an $i\in[n]$ such that for any $v\in Q_n$, the sum of the Hamming distances from $v$ to $C(0,i)$ and $C(1,i)$ is at most $2R+1$. We newly define what it means for an asymmetric covering code to be normal, and consider the worst case…
Let $k$ be a nonnegative integer, and let $α$ and $β$ be two permutations of $n$ symbols. We say that $α$ and $β$ $k$-commute if $H(αβ, βα)=k$, where $H$ denotes the Hamming metric between permutations. In this paper, we consider the problem of finding the permutations that $k$-commute with a given permutation. Our main result is a characterization of permutations that $k$-commute with a given permutation $β$ in terms of blocks in cycles in the decomposition of $β$ as a product of disjoint cycles. Using this characterization, we provide formulas for the number of permutations that $k$-commute …
An $(n,k)$ sequence covering array is a set of permutations of $[n]$ such that each sequence of $k$ distinct elements of $[n]$ is a subsequence of at least one of the permutations. An $(n,k)$ sequence covering array is perfect if there is a positive integer $λ$ such that each sequence of $k$ distinct elements of $[n]$ is a subsequence of precisely $λ$ of the permutations. While relatively close upper and lower bounds for the minimum size of a sequence covering array are known, this is not the case for perfect sequence covering arrays. Here we present new nontrivial bounds for the latter. In pa…
A notion of $t$-designs in the symmetric group on $n$ letters was introduced by Godsil in 1988. In particular $t$-transitive sets of permutations form a $t$-design. We derive upper bounds on the covering radius of these designs, as a function of $n$ and $t$ and in terms of the largest zeros of Charlier polynomials.
Given positive integers $n$ and $d$, let $M(n,d)$ denote the maximum size of a permutation code of length $n$ and minimum Hamming distance $d$. The Gilbert-Varshamov bound asserts that $M(n,d) \geq n!/V(n,d-1)$ where $V(n,d)$ is the volume of a Hamming sphere of radius $d$ in $§_n$. Recently, Gao, Yang, and Ge showed that this bound can be improved by a factor $Ω(\log n)$, when $d$ is fixed and $n \to \infty$. Herein, we consider the situation where the ratio $d/n$ is fixed and improve the Gilbert-Varshamov bound by a factor that is \emph{linear in $n$}. That is, we show that if $d/n < 0.5$, t…
An (n,d)-permutation code is a subset C of Sym(n) such that the Hamming distance d_H between any two distinct elements of C is at least equal to d. In this paper, we use the characterisation of the isometry group of the metric space (Sym(n),d_H) in order to develop generating algorithms with rejection of isomorphic objects. To classify the (n,d)-permutation codes up to isometry, we construct invariants and study their efficiency. We give the numbers of non-isometric (4,3)- and (5,4)- permutation codes. Maximal and balanced (n,d)-permutation codes are enumerated in a constructive way.
A $λ$-fold $r$-packing (multiple radius-$r$ covering) in a Hamming metric space is a code $C$ such that the radius-$r$ balls centered in $C$ cover each vertex of the space by not more (not less, respectively) than $λ$ times. The well-known $r$-error-correcting codes correspond to the case $λ=1$, while in general multifold $r$-packing are related with list decodable codes. We (a) propose asymptotic bounds for the maximum size of a $q$-ary $2$-fold $1$-packing as $q$ grows; (b) prove that a $q$-ary distance-$2$ MDS code of length $n$ is an optimal $n$-fold $1$-packing if $q\ge 2n$; (c) derive an…
A PSCA$(v, t, λ)$ is a multiset of permutations of the $v$-element alphabet $\{0, \dots, v-1\}$ such that every sequence of $t$ distinct elements of the alphabet appears in the specified order in exactly $λ$ permutations. For $v \geq t$, let $g(v, t)$ be the smallest positive integer $λ$ such that a PSCA$(v, t, λ)$ exists. We present an explicit construction that proves $g(v,t) = O(v^{t(t-2)})$ for fixed $t \geq 4$. The method of construction involves taking a permutation representation of the group of projectivities of a suitable projective space of dimension $t - 2$ and deleting all but a ce…
We denote by $\mathbb{F}_q$ the field with $q$ elements. A radius-$r$ extended ball with center in a $1$-dimensional vector subspace $V$ of $\mathbb{F}_q^3$ is the set of elements of $\mathbb{F}_q^3$ with Hamming distance to $V$ at most $r$. We define $c(q)$ as the size of a minimum covering of $\fqt$ by radius-$1$ extended balls. We define a semiqueen as a piece of a toroidal chessboard that extends the covering range of a rook by the southwest-northeast diagonal containing it. Let $ξ_D(n)$ be the minimum number of semiqueens of the $n\times n$ toroidal board necessary to cover the entire boa…
This work shows several direct and recursive constructions of ordered covering arrays using projection, fusion, column augmentation, derivation, concatenation and cartesian product. Upper bounds on covering codes in NRT spaces are also obtained by improving a general upper bound. We explore the connection between ordered covering arrays and covering codes in NRT spaces, which generalize similar results for the Hamming metric. Combining the new upper bounds for covering codes in NRT spaces and ordered covering arrays, we improve upper bounds on covering codes in NRT spaces for larger alphabets.…
Let L be a Desarguesian 2-spread in the Grassmann graph $J_q(n,2)$. We prove that the collection of the 4-subspaces, which do not contain subspaces from L is a completely regular code in $J_q(n,4)$. Similarly, we construct a completely regular code in the Johnson graph $J(n,6)$ from the Steiner quadruple system of the extended Hamming code. We obtain several new completely regular codes covering radius 1 in the Grassmann graph $J_2(6,3)$ using binary linear programming.
In this paper, we introduce a new parameter of a code, referred to as the remoteness, which can be viewed as a dual to the covering radius. Indeed, the remoteness is the minimum radius needed for a single ball to cover all codewords. After giving some general results about the remoteness, we then focus on the remoteness of permutation codes. We first derive upper and lower bounds on the minimum cardinality of a code with a given remoteness. We then study the remoteness of permutation groups. We show that the remoteness of transitive groups can only take two values, and we determine the remoten…
A covering code is a subset $\mathcal{C} \subseteq \{0,1\}^n$ with the property that any $z \in \{0,1\}^n$ is close to some $c \in \mathcal{C}$ in Hamming distance. For every $ε,δ>0$, we show a construction of a family of codes with relative covering radius $δ+ ε$ and rate $1 - \mathrm{H}(δ) $ with block length at most $\exp(O((1/ε) \log (1/ε)))$ for every $ε> 0$. This improves upon a folklore construction which only guaranteed codes of block length $\exp(1/ε^2)$. The main idea behind this proof is to find a distribution on codes with relatively small support such that most of these codes have…
Consider the family of all perfect matchings of the complete graph $K_{2n}$ with $2n$ vertices. Given any collection $\mathcal M$ of perfect matchings of size $s$, there exists a maximum number $f(n,x)$ such that if $s\leq f(n,x)$, then there exists a perfect matching that agrees with each perfect matching in $\mathcal M$ in at most $x-1$ edges. We use probabilistic arguments to give several lower bounds for $f(n,x)$. We also apply the Lovász local lemma to find a function $g(n,x)$ such that if each edge appears at most $g(n, x)$ times then there exists a perfect matching that agrees with each…
On the Generalized Packing and Covering Radii of Codes
Institute of Mathematics and Interdisciplinary Sciences, Xidian University（西安电子科技大学数学与交叉科学研究院）；Department of Electrical and Computer Engineering, McMaster University（麦克马斯特大学电气与计算机工程系）
We consider rational functions of the form $V(x)/U(x)$, where both $V(x)$ and $U(x)$ are polynomials over the finite field $\mathbb{F}_q$. Polynomials that permute the elements of a field, called {\it permutation polynomials ($PPs$)}, have been the subject of research for decades. Let ${\mathcal P}^1(\mathbb{F}_q)$ denote $\mathbb{Z}_q \cup \{\infty\}$. If the rational function, $V(x)/U(x)$, permutes the elements of ${\mathcal P}^1(\mathbb{F}_q)$, it is called a {\em permutation rational function (PRf)}. Let $N_d(q)$ denote the number of PPs of degree $d$ over $\mathbb{F}_q$, and let $N_{v,u}(…
We prove the conjecture of Baik, Deift, and Johansson which says that with respect to the Plancherel measure on the set of partitions of $n$, the 1st, 2nd, and so on, rows behave, suitably scaled, like the 1st, 2nd, and so on, eigenvalues of a Gaussian random Hermitian matrix as $n$ goes to infinity. Our proof is based on an interplay between maps on surfaces and ramified coverings of the sphere. We also establish a connection of this problem with intersection theory on the moduli spaces of curves.
A permutation array $A$ is a set of permutations on a finite set $Ω$, say of size $n$. Given distinct permutations $π, σ\in Ω$, we let $hd(π, σ) = |\{ x\in Ω: π(x) \ne σ(x) \}|$, called the Hamming distance between $π$ and $σ$. Now let $hd(A) =$ min$\{ hd(π, σ): π, σ\in A \}$. For positive integers $n$ and $d$ with $d\le n$, we let $M(n,d)$ be the maximum number of permutations in any array $A$ satisfying $hd(A) \geq d$. There is an extensive literature on the function $M(n,d)$, motivated in part by suggested applications to error correcting codes for message transmission over power lines. A b…
We study the size (or volume) of balls in the metric space of permutations, $S_n$, under the infinity metric. We focus on the regime of balls with radius $r = ρ\cdot (n\!-\!1)$, $ρ\in [0,1]$, i.e., a radius that is a constant fraction of the maximum possible distance. We provide new lower bounds on the size of such balls. These new lower bounds reduce the asymptotic gap to the known upper bounds to at most $0.029$ bits per symbol. Additionally, they imply an improved ball-packing bound for error-correcting codes, and an improved upper bound on the size of optimal covering codes.
