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• W2964005479 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." @default.
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• W2964005479 date "2014-09-01" @default.
• W2964005479 modified "2023-10-10" @default.
• W2964005479 title "Covering Sets for Limited-Magnitude Errors" @default.
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• W2964005479 doi "https://doi.org/10.1109/tit.2014.2338078" @default.
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