Covering Sets for Limited-Magnitude Errors

Abstract

For a set M = {-μ, -μ + 1, ... , λ} \ {0} with nonnegative integers λ, μ <; q not both 0, a subset S of the residue class ring Z <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> modulo an integer q > 1 is called a (λ, μ; q)-covering set if MS = {ms mod q : m ∈ M, s ∈ S} = Z <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> . Small covering sets play an important role in codes correcting limited-magnitude errors. We give an explicit construction of a (λ, μ; q)-covering set S, which is of the size q <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1+o(1)</sup> max{λ, μ} <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-1/2</sup> for almost all integers q ≥ 1 and optimal order of magnitude (that is up to a multiplicative constant) p max{λ, μ} <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-1</sup> if q = p is prime. Furthermore, using a bound on the fourth moment of character sums of Cochrane and Shi that there is a (λ, μ; q)-covering set of size at most q <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1+o(1)</sup> max{λ, μ} <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-1/2</sup> for any integer q ≥ 1, however the proof of this bound is not constructive.