28 research outputs found

    On subsets of the hypercube with prescribed Hamming distances

    Get PDF
    A celebrated theorem of Kleitman in extremal combinatorics states that a collection of binary vectors in {0,1}^n with diameter d has cardinality at most that of a Hamming ball of radius d/2. In this paper, we give an algebraic proof of Kleitman's Theorem, by carefully choosing a pseudo-adjacency matrix for certain Hamming graphs, and applying the Cvetković bound on independence numbers. This method also allows us to prove several extensions and generalizations of Kleitman's Theorem to other allowed distance sets, in particular blocks of consecutive integers of width much smaller than n. We also improve on a theorem of Alon about subsets of F^n_p whose difference set does not intersect {0,1}^n nontrivially

    On subsets of the hypercube with prescribed Hamming distances

    Get PDF
    A celebrated theorem of Kleitman in extremal combinatorics states that a collection of binary vectors in {0,1}^n with diameter d has cardinality at most that of a Hamming ball of radius d/2. In this paper, we give an algebraic proof of Kleitman's Theorem, by carefully choosing a pseudo-adjacency matrix for certain Hamming graphs, and applying the Cvetković bound on independence numbers. This method also allows us to prove several extensions and generalizations of Kleitman's Theorem to other allowed distance sets, in particular blocks of consecutive integers of width much smaller than n. We also improve on a theorem of Alon about subsets of F^n_p whose difference set does not intersect {0,1}^n nontrivially

    On Optimal Anticodes over Permutations with the Infinity Norm

    Full text link
    Motivated by the set-antiset method for codes over permutations under the infinity norm, we study anticodes under this metric. For half of the parameter range we classify all the optimal anticodes, which is equivalent to finding the maximum permanent of certain (0,1)(0,1)-matrices. For the rest of the cases we show constraints on the structure of optimal anticodes

    On Robust Colorings of Hamming-Distance Graphs

    Get PDF
    Hq(n, d) is defined as the graph with vertex set Znq and where two vertices are adjacent if their Hamming distance is at least d. The chromatic number of these graphs is presented for various sets of parameters (q, n, d). For the 4-colorings of the graphs H2(n, n − 1) a notion of robustness is introduced. It is based on the tolerance of swapping colors along an edge without destroying properness of the coloring. An explicit description of the maximally robust 4-colorings of H2 (n, n − 1) is presented

    Colorings of Hamming-Distance Graphs

    Get PDF
    Hamming-distance graphs arise naturally in the study of error-correcting codes and have been utilized by several authors to provide new proofs for (and in some cases improve) known bounds on the size of block codes. We study various standard graph properties of the Hamming-distance graphs with special emphasis placed on the chromatic number. A notion of robustness is defined for colorings of these graphs based on the tolerance of swapping colors along an edge without destroying the properness of the coloring, and a complete characterization of the maximally robust colorings is given for certain parameters. Additionally, explorations are made into subgraph structures whose identification may be useful in determining the chromatic number

    Combinatorics

    Get PDF
    Combinatorics is a fundamental mathematical discipline that focuses on the study of discrete objects and their properties. The present workshop featured research in such diverse areas as Extremal, Probabilistic and Algebraic Combinatorics, Graph Theory, Discrete Geometry, Combinatorial Optimization, Theory of Computation and Statistical Mechanics. It provided current accounts of exciting developments and challenges in these fields and a stimulating venue for a variety of fruitful interactions. This is a report on the meeting, containing extended abstracts of the presentations and a summary of the problem session
    corecore