Shaofang Hong

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Determining deep holes is an important open problem in decoding ReedSolomon codes. It is well known that the received word is trivially a deep hole if the degree of its Lagrange interpolation polynomial equals the dimension of the ReedSolomon code. For the standard Reed-Solomon codes [p − 1, k]p with p a prime, Cheng and Murray conjectured in 2007 that(More)
Determining deep holes is an important topic in decoding Reed-Solomon codes. In a previous paper [8], we showed that the received word u is a deep hole of the standard Reed-Solomon codes [q− 1, k]q if its Lagrange interpolation polynomial is the sum of monomial of degree q− 2 and a polynomial of degree at most k− 1. In this paper, we extend this result by(More)
Let k ≥ 0, a ≥ 1 and b ≥ 0 be integers. We define the arithmetical function gk,a,b for any positive integer n by gk,a,b(n) := (b+na)(b+(n+1)a)···(b+(n+k)a) lcm(b+na,b+(n+1)a,··· ,b+(n+k)a) . Letting a = 1 and b = 0, then gk,a,b becomes the arithmetical function introduced previously by Farhi. Farhi proved gk,1,0 is periodical and k! is a period. Hong and(More)
For relatively prime positive integers u0 and r, we consider the arithmetic progression {uk := u0 + kr} n k=0 . Define Ln := lcm{u0, u1, . . . , un} and let a ≥ 2 be any integer. In this paper, we show that, for integers α, r ≥ a and n ≥ 2αr, we have Ln ≥ u0r (r + 1). In particular, letting a = 2 yields an improvement to the best previous lower bound on Ln(More)
Let p be a prime. We obtain good bounds for the p-adic sizes of the coefficients of the divided universal Bernoulli number B̂n n when n is divisible by p− 1. As an application, we give a simple proof of Clarke’s 1989 universal von Staudt theorem. We also establish the universal Kummer congruences modulo p for the divided universal Bernoulli numbers for the(More)
In this paper, we investigate the 2-adic valuation of the Stirling numbers S(n, k) of the second kind. We show that v2(S(2n + 1, k + 1)) = s2(n) − 1 for any positive integer n, where s2(n) is the sum of binary digits of n. This confirms a conjecture of Amdeberhan, Manna and Moll. We show also that v2(S(4i, 5)) = v2(S(4i + 3, 5)) if and only if i 6≡ 7 (mod(More)