Gretchen L. Matthews

Weierstrass Pairs and Minimum Distance of Goppa Codes

We prove that elements of the Weierstrassgap set of a pair of points may be used to define a geometricGoppa code which has minimum distance greater than the usuallower bound. We determine the Weierstrass gap set of a pair ofany two Weierstrass points on a Hermitian curve and use thisto increase the lower bound on the minimum distance of particularcodes defined using a linear combination of the two points.

Codes from the Suzuki function field

We construct algebraic geometry (AG) codes from the function field F(2/sup 2n+1/)(x,y)/F(2/sup 2n+1/) defined by y(2/sup 2n+1/)-y=(x(2/sup 2n+/)-x) where n is a positive integer. These codes are supported by two places, and many have parameters that are better than those of any comparable code supported by one place of the same function field. To define such codes, we determine and exploit the structure of the Weierstrass gap set of an arbitrary pair of rational places of F(2/sup 2n+1/)(x,y)/F(2/sup 2n+1/). Moreover, we find some codes over F/sub 8/ with parameters that are better than any known code.

The Weierstrass Semigroup of an m-tuple of Collinear Points on a Hermitian Curve

We examine the structure of the Weierstrass semigroup of an m-tuple of points on a smooth, projective, absolutely irreducible curve X over a finite field \(\mathbb{F}\). A criteria is given for determining a minimal subset of semigroup elements which generate such a semigroup where 2 \( \leq m \leq |\mathbb{F}|\). For all 2 mq + 1, we determine the Weierstrass semigroup of any m-tuple of collinear \(\mathbb{F}_{q^2}\)-rational points on a Hermitian curve y q + y = x q + 1.

Acyclic colorings of products of trees

We obtain bounds for the coloring numbers of products of trees for three closely related types of colorings: acyclic, distance 2, and L(2,1).

On Numerical Semigroups Generated by Generalized Arithmetic Sequences

Abstract Given a numerical semigroup S, let M(S) = S\{0} and (lM(S) − lM(S)) = {x ∈ ℕ0 : x + lM(S) ⊆ lM(S)}. Define associated numerical semigroups B(S) ≔ (M(S) − M(S)) and . Set B 0(S) = S, and for i ≥ 1, define B i (S) ≔ B(B i−1(S)). Similarly, set L 0(S) = S, and for i ≥ 1, define L i (S) ≔ L(L i−1(S)). These constructions define two finite ascending chains of numerical semigroups S = B 0(S) ⊆ B 1(S) ⊆ … ⊆ B β(S)(S) = ℕ0 and S = L 0(S) ⊆ L 1(S) ⊆ … ⊆ L λ(S)(S) = ℕ0. It has been shown that not all numerical semigroups S have the property that B i (S) ⊆ L i (S) for all i ≥ 0. In this paper, we prove that if S is a numerical semigroup with a set of generators that form a generalized arithmetic sequence, then B i (S) ⊆ L i (S) for all i ≥ 0. Moreover, we see that this containment is not necessarily satisfied if a set of generators of S form an almost arithmetic sequence. In addition, we characterize numerical semigroups generated by generalized arithmetic sequences that satisfy other semigroup properties, such as symmetric, pseudo-symmetric, and Arf.

On the floor and the ceiling of a divisor

Given a divisor A of a function field, there is a unique divisor of minimum degree that defines the same vector space of rational functions as A and there is a unique divisor of maximum degree that defines the same vector space of rational differentials as A. These divisors are called the floor and the ceiling of A. A method is given for finding both the floor and the ceiling of a divisor. The floor and the ceiling of a divisor give new bounds for the minimum distance of algebraic geometry codes. The floor and the ceiling of a divisor supported by collinear places of the Hermitian function field are determined. Finally, we find the exact code parameters for a large class of algebraic geometry codes constructed from the Hermitian function field.

One-point codes using places of higher degree

In IEEE Transactions on Information Theory , vol. 48, no. 2, pp. 535-537, Feb. 2002, Xing and Chen show that there exist algebraic-geometry (AG) codes from the Hermitian function field over F/sub q//sup 2/ constructed using F/sub q//sup 2/-rational divisors which are improvements over the much-studied one-point Hermitian codes. In this correspondence, we construct such codes by using a place P of degree r > 1. This motivates a study of gap numbers and pole numbers at places of higher degree. In fact, the code parameters are estimated using the Weierstrass gap set of the place P and relating it to the gap set of the r-tuple of places of degree one lying over P in a constant field extension of degree r.

On the Acyclic Chromatic Number of Hamming Graphs

An acyclic coloring of a graph G is a proper coloring of the vertex set of G such that G contains no bichromatic cycles. The acyclic chromatic number of a graph G is the minimum number k such that G has an acyclic coloring with k colors. In this paper, acyclic colorings of Hamming graphs, products of complete graphs, are considered. Upper and lower bounds on the acyclic chromatic number of Hamming graphs are given.

Lifting the Fundamental Cone and Enumerating the Pseudocodewords of a Parity-Check Code

The performance of message-passing iterative decoding and linear programming decoding depends on the Tanner graph representation of the code. If the underlying graph contains cycles, then such algorithms could produce a noncodeword output. The study of pseudocodewords aims to explain this noncodeword output. We examine the structure of the pseudocodewords and show that there is a one-to-one correspondence between graph cover pseudocodewords and integer points in a lifted fundamental cone. This gives a simple proof that the generating function of the pseudocodewords for a general parity-check code is rational (a fact first proved by Li, Lu, and Wang (Lecture Notes in Computer Science, vol. 5557, 2009) via other methods). Our approach yields algorithms for producing this generating function and provides tools for studying the irreducible pseudocodewords. Specifically, Barvinoks algorithm and the Barvinok-Woods projection algorithm are applied, and irreducible pseudocodewords are found via a Hilbert basis for the lifted fundamental cone.

A variant of the Frobenius problem and generalized Suzuki semigroups

Given relatively prime positive integers a1, . . . , ak, let S denote the set of all linear combinations of a1, . . . , ak with nonnegative integral coefficients. The Frobenius problem is to determine the largest integer g(S) which is not representable as such a linear combination. A related question is to determine the set B(S) of integers x that are representable as differences x = s1 − a1 = . . . = sk − ak for some si ∈ S. The construction B(S) can be iterated to obtain a chain of numerical semigroups. We compare this chain to the one obtained by iterating the Lipman semigroup construction. In particular, we consider these chains for generalized Suzuki semigroups.