2017 CMS Winter Meeting

Waterloo, December 8 - 11, 2017

Design Theory
Org: Hadi Kharaghani (University of Lethbridge) and Doug Stinson (University of Waterloo)

PETER DANZINGER, Ryerson

ILIAS KOTSIREAS, Wilfred Laurier

DON KREHER, Michigan Technological

PETR LISONEK, Simon Fraser

LUCIA MOURA, Ottawa

MIKE NEWMAN, Ottawa

DANIEL PANARIO, Carleton University
Covering arrays from m-sequences and character sums over finite fields  [PDF]

A covering array of strength $t$ on $v$ symbols is an array with the property that, for every $t$-combination of column vectors, every one of the possible $v^t$ $t$-tuples of symbols appears as a row at least once in the subarray defined by these column vectors. Arrays whose rows are cyclic shifts of an m-sequence over a finite field possess many combinatorial properties and have been used to construct various combinatorial objects; see [2].

In this talk we consider covering arrays consisting of discrete logarithms of carefully selected m-sequence elements. Inspired by [1], we connect the covering array definition for this type of arrays to the value of certain character sums over finite fields. Taking advantage of the balanced way in which the m-sequence elements are distributed, we are able to evaluate these sums. This provides new infinite families of covering arrays of arbitrary strength [3].

Joint work with L. Moura, B. Stevens and G. Tzanakis.

References:

[1] C.J. Colbourn, Covering arrays from cyclotomy, Designs, Codes and Cryptography 55 (2010), 201-219.

[2] L. Moura, G. L. Mullen, D. Panario. Finite field constructions of combinatorial arrays, Designs, Codes and Cryptography 78 (2016), 197-219.

[3] G. Tzanakis, L. Moura, D. Panario, B. Stevens. Covering arrays from m-sequences and character sums, Designs, Codes and Cryptography 85 (2017), 437-456.

DAVID PIKE, Memorial

DOUG STINSON, Waterloo

STEVE WANG, Carleton

QING XIANG, Delaware