On the Modulus in Matching Vector Codes
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A k-query locally decodable code (LDC) C allows one to encode any n-symbol message x as a codeword C(x) of N symbols such that each symbol of x can be recovered by looking at k symbols of C(x), even if a constant fraction of C(x) have been corrupted. Currently, the best known LDCs are matching vector codes (MVCs). A modulus m=p_1^α_1p_2^α_2⋯ p_r^α_r may result in an MVC with k≤ 2^r and N=exp(exp(O((log n)^1-1/r (loglog n)^1/r))). The m is good if it is possible to have k<2^r. The good numbers yield more efficient MVCs. Prior to this work, there are only finitely many good numbers. All of them were obtained via computer search and have the form m=p_1p_2. In this paper, we study good numbers of the form m=p_1^α_1p_2^α_2. We show that if m=p_1^α_1p_2^α_2 is good, then any multiple of m of the form p_1^β_1p_2^β_2 must be good as well. Given a good number m=p_1^α_1p_2^α_2, we show an explicit method of obtaining smaller good numbers that have the same prime divisors. Our approach yields infinitely many new good numbers.
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