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Determining deep holes is an important open problem in decoding Reed-Solomon 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 Reed-Solomon code. For the standard Reed-Solomon codes [p − 1, k]p with p a prime, Cheng and Murray conjectured in 2007 that… (More)

- SHAOFANG HONG
- 2012

Let F q be the finite field of q elements with characteristic p and F q m its extension of degree m. Fix a nontrivial additive character ψ of F p. For any Laurent polynomial −1 n ], we form the exponential sum S * m (f) := The corresponding L-function L * (f, t) is defined by L * (f, t) := exp (∞ ∑ m=0 S * m (f) t m m). The corresponding L-function L(f, t)… (More)

- SHAOFANG HONG
- 2009

For relatively prime positive integers u 0 and r, we consider the arithmetic progression {u k := u 0 + kr} n k=0. Define Ln := lcm{u 0 , u 1 ,. .. , 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 ≥ u 0 r α+a−2 (r + 1) n. In particular, letting a = 2 yields an improvement to the best previous… (More)

- SHAOFANG HONG, GUOYOU QIAN
- 2009

Let k ≥ 0, a ≥ 1 and b ≥ 0 be integers. We define the arithmetical function g k,a,b for any positive integer n by g k,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 g k,a,b becomes the arithmetical function introduced previously by Farhi. Farhi proved g k,1,0 is periodical and k! is a period. divides… (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… (More)

Let p be a prime. We obtain good bounds for the p-adic sizes of the coefficients of the divided universal Bernoulli numberˆB n 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)