arXiv subjects

Jiyou Li

Publications and source records attributed to Jiyou Li.

23 records · Page 2Linked to original sources

arXiv2013

On Determining Deep Holes of Generalized Reed-Solomon Codes

For a linear code, deep holes are defined to be vectors that are further away from codewords than all other vectors. The problem of deciding whether a received word is a deep hole for generalized Reed-Solomon codes is proved to be co-NP-complete. For the extended Reed-Solomon codes $RS_q(\F_q,k)$, a conjecture was made to classify deep holes by Cheng and Murray in 2007. Since then a lot of effort has been made to prove the conjecture, or its various forms. In this paper, we classify deep holes completely for generalized Reed-Solomon codes $RS_p (D,k)$, where $p$ is a prime, $|D| > k \geqslant \frac{p-1}{2}$. Our techniques are built on the idea of deep hole trees, and several results concerning the Erd{ö}s-Heilbronn conjecture.

cs.IT↗

arXiv2012

On the Odlyzko-Stanley enumeration problem and Waring's problem over finite fields

We obtain an asymptotic formula on the Odlyzko-Stanley enumeration problem. Let $N_m^*(k,b)$ be the number of $k$-subsets $S\subseteq F_p^*$ such that $\sum_{x\in S}x^m=b$. If $m 0$ such that | N_m^*(k,b)-p^{-1}{p-1 \choose k}|\leq {p^{1-ε}+mk-m \choose k}. In addition, let $γ'(m,p)$ denote the distinct Waring's number $(\mod p)$, the smallest positive integer $k$ such that every integer is a sum of m-th powers of $k$-distinct elements $(\mod p)$. The above bound implies that there is a constant $ε(δ)>0$ such for any prime $p$ and any $m<p^{1-δ}$, if $ε^{-1}<(e-1)p^{δ-ε}$, then $$γ'(m,p)\leq ε^{-1}.$$

math.NT↗

arXiv2011

On the average sensitivity of laced Boolean functions

In this paper we obtain the average sensitivity of the laced Boolean functions. This confirms a conjecture of Shparlinski. We also compute the weights of the laced Boolean functions and show that they are almost balanced.

cs.IT↗

arXiv2010

Permutation Polynomials of Degree 6 or 7 over Finite Fields of Characteristic 2

In \cite{D1}, Dickson listed all permutation polynomials up to degree 5 over an arbitrary finite field, and all permutation polynomials of degree 6 over finite fields of odd characteristic. The classification of degree 6 permutation polynomials over finite fields of characteristic 2 was left incomplete. In this paper we complete the classification of permutation polynomials of degree 6 over finite fields of characteristic 2. In addition, all permutation polynomials of degree 7 over finite fields of characteristic 2 are classified.

math.CO↗

arXiv2007

On the subset sum problem over finite fields

The subset sum problem over finite fields is a well-known {\bf NP}-complete problem. It arises naturally from decoding generalized Reed-Solomon codes. In this paper, we study the number of solutions of the subset sum problem from a mathematical point of view. In several interesting cases, we obtain explicit or asymptotic formulas for the solution number. As a consequence, we obtain some results on the decoding problem of Reed-Solomon codes.

math.NT↗