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Gary Mcguire

  1. There is no 16-Clue Sudoku: Solving the Sudoku Minimum Number of Clues Problem.

    Thu, 01/05/2012 - 15:01 — SomNambul
    Authors: Gary Mcguire, Bastian Tugemann, Gilles Civario
    Subjects: Data Structures and Algorithms
    Abstract

    We apply our new hitting set enumeration algorithm to solve the sudoku
    minimum number of clues problem, which is the following question: What is the
    smallest number of clues (givens) that a sudoku puzzle may have? It was
    conjectured that the answer is 17. We have performed an exhaustive search for a
    16-clue sudoku puzzle, and we did not find one, thereby proving that the answer
    is indeed 17. This article describes our method and the actual search.

  2. Characteristic Polynomial of Supersingular Abelian Varieties over Finite Fields.

    Fri, 10/07/2011 - 15:01 — SomNambul
    Authors: Gary Mcguire, Vijaykumar Singh, Alexey Zaytsev
    Subjects: Algebraic Geometry
    Abstract

    In this article, we give a complete description of the characteristic
    polynomials of supersingular abelian varieties over finite fields. We list them
    for the dimensions upto 7.

  3. The Intersection of Two Fermat Hypersurfaces in P^3 via Computation of Quotient Curves.

    Mon, 11/23/2009 - 16:01 — SomNambul
    Authors: Gary Mcguire, Vijaykumar Singh
    Subjects: Algebraic Geometry
    Abstract

    We study the intersection of two particular Fermat hypersurfaces in
    $\mathbb{P}^3$ over a finite field.

    Using the Kani-Rosen decomposition we study arithmetic properties of this
    curve in terms of its quotients. Explicit computation of the quotients is done
    using a Gr\"obner basis algorithm. We also study the $p$-rank, zeta function
    and number of rational points. We make a conjecture about the Jacobian of the
    genus 2 quotient, which we show is $(4,4)$-split.

  4. Proof of a Conjecture on the Sequence of Exceptional Numbers, Classifying Cyclic Codes and APN Functions.

    Mon, 09/21/2009 - 13:59 — SomNambul
    Authors: Gary Mcguire, Fernando Hernando
    Subjects: Information Theory
    Abstract

    We prove a conjecture that classifies exceptional numbers. This conjecture
    arises in two different ways, from cryptography and from coding theory. An odd
    integer $t\geq 3$ is said to be exceptional if $f(x)=x^t$ is APN (Almost
    Perfect Nonlinear) over $\mathbb{F}_{2^n}$ for infinitely many values of $n$.
    Equivalently, $t$ is exceptional if the binary cyclic code of length $2^n-1$
    with two zeros $\omega, \omega^t$ has minimum distance 5 for infinitely many
    values of $n$. The conjecture we prove states that every exceptional number has
    the form $2^i+1$ or $4^i-2^i+1$.

  5. A few more functions that are not APN infinitely often.

    Tue, 09/15/2009 - 11:27 — SomNambul
    Authors: Yves Aubry, Gary Mcguire, François Rodier
    Subjects: Algebraic Geometry
    Abstract

    We consider exceptional APN functions on ${\bf F}_{2^m}$, which by definition
    are functions that are not APN on infinitely many extensions of ${\bf
    F}_{2^m}$. Our main result is that polynomial functions of odd degree are not
    exceptional, provided the degree is not a Gold member ($2^k+1$) or a
    Kasami-Welch number ($4^k-2^k+1$). We also have partial results on functions of
    even degree, and functions that have degree $2^k+1$.

Math-arch

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