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Register a new userCovering radius in the Hamming permutation space
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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 $Ssubseteq mathcal{S}_n$ with the property that for any $rhoin mathcal{S}_n$ there exists some $sigmain S$ such that the Hamming distance between $rho$ and $sigma$ is at most $n-s$. The value of $f(n,2)$ is the subject of a conjecture by Kezdy and Snevily, which implies several famous conjectures about latin squares. We prove that the odd $n$ case of the Kezdy-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 $1leq sleq n-2$ and that [f(n,s)>leftlfloor frac{2+sqrt{2s-2}}{2}rightrfloor frac{n}{2}] if $sgeq 3$.
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In this paper we study properties and invariants of matrix codes endowed with the rank metric, and relate them to the covering radius. We introduce new tools for the analysis of rank-metric codes, such as puncturing and shortening constructions. We give upper bounds on the covering radius of a code by applying different combinatorial methods. We apply the various bounds to the classes of maximal rank distance and quasi maximal rank distance codes.
We describe the adjacency of vertices of the (unbounded version of the) set covering polyhedron, in a similar way to the description given by Chvatal for the stable set polytope. We find a sufficient condition for adjacency, and characterize it with similar conditions in the case where the underlying matrix is row circular. We apply our findings to show a new infinite family of minimally nonideal matrices.
Following Britz, Johnsen, Mayhew and Shiromoto, we consider demi-ma-troids as a(nother) natural generalization of matroids. As they have shown, demi-ma-troids are the appropriate combinatorial objects for studying Weis duality. Our results here apport further evidence about the trueness of that observation. We define the Hamming polynomial of a demimatroid $M$, denoted by $W(x,y,t)$, as a generalization of the extended Hamming weight enumerator of a matroid. The polynomial $W(x,y,t)$ is a specialization of the Tutte polynomial of $M$, and actually is equivalent to it. Guided by work of Johnsen, Roksvold and Verdure for matroids, we prove that Betti numbers of a demimatroid and its elongations determine the Hamming polynomial. Our results may be applied to simplicial complexes since in a canonical way they can be viewed as demimatroids. Furthermore, following work of Brylawski and Gordon, we show how demimatroids may be generalized one step further, to combinatroids. A combinatroid, or Brylawski structure, is an integer valued function $rho$, defined over the power set of a finite ground set, satisfying the only condition $rho(emptyset)=0$. Even in this extreme generality, we will show that many concepts and invariants in coding theory can be carried on directly to combinatroids, say, Tutte polynomial, characteristic polynomial, MacWilliams identity, extended Hamming polynomial, and the $r$-th generalized Hamming polynomial; this last one, at least conjecturelly, guided by the work of Jurrius and Pellikaan for linear codes. All this largely extends the notions of deletion, contraction, duality and codes to non-matroidal structures.
The Norton product is defined on each eigenspace of a distance regular graph by the orthogonal projection of the entry-wise product. The resulting algebra, known as the Norton algebra, is a commutative nonassociative algebra that is useful in group theory due to its interesting automorphism group. We provide a formula for the Norton product on each eigenspace of a Hamming graph using linear characters. We construct a large subgroup of automorphisms of the Norton algebra of a Hamming graph and completely describe the automorphism group in some cases. We also show that the Norton product on each eigenspace of a Hamming graph is as nonassociative as possible, except for some special cases in which it is either associative or equally as nonassociative as the so-called double minus operation previously studied by the author, Mickey, and Xu. Our results restrict to the hypercubes and extend to the halved and/or folded cubes, the bilinear forms graphs, and more generally, all Cayley graphs of finite abelian groups.
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