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| 1 | On deep holes of standard Reed-Solomon codes显示文摘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 there is no other deep holes except the trivial ones. In this paper, we show that this conjecture is not true. In fact, we find a new class of deep holes for standard Reed-Solomon codes [q-1, k]q with q a power of the prime p. Let q≥4 and 2≤k≤q-2. We show that the received word u is a deep hole if its Lagrange interpolation polynomial is the sum of monomial of degree q-2 and a polynomial of degree at most k-1. So there are at least 2(q-1)qk deep holes if k q-3. | WU RongJun HONG ShaoFang | 2012 | Science China Mathematics2012，55，12： | 9 |
| 2 | Additive characters and orthogonal systems of polynomials in several indeterminates over residue class rings显示文摘Let m, n and k be positive integers, m>1 and k≤n . Let Z m =Z/ m Z denote the residue class ring modulo m . Let Z denote the group of additive characters of ring Z m , x 0 denote the identity of Z . A necessary and sufficient condition for the system f\-1, ..., f\-k∈ Z m [ x\-1, ..., x\-n ] to be orthogonal is given: for all additive characters ψ 1, ..., ψ k ∈Z with ψ i≠x 0 for some i, 1≤i≤k, ∑(c\-1, ..., c\-n)∈ Z n m ψ(f\-1(c\-1, ..., c\-n))...ψ k(f\-k(c\-1, ..., c\-n))=0. As corollaries, the result of Shiue_Sun_Zhang and the result of Sun are obtained. | HONG Shaofang Department of Mathematics, Sichuan University, Chengdu 610064, China | 1998 | Chinese Science Bulletin1998，43，4： | 3 |
| 3 | A novel electronic tongue combined MLAPS with stripping voltammetry for environment detection显示文摘 | Men Hong Zou Shaofang Li Yi | 2005 | Sensors and Actuators B2005，110，： | 1 |
| 4 | On the divisibility of power LCM matrices by power GCD matrices显示文摘 | Jianrong Zhao Shaofang Hong Qunying Liao K. P. Shum | 2007 | Czechoslovak Mathematical Journal2007，，1： | 1 |
| 5 | Determinants of Matrices Associated with Incidence Functions on Posets显示文摘 | Shaofang Hong Qi Sun | 2004 | Czechoslovak Mathematical Journal2004，，2： | 1 |
| 6 | A novel electronic tongue combined MLAPS with stripping voltammetry for environmental detection 显示文摘 | Hong Men Shaofang Zou Yi Li | 2005 | Sensors and Actuators B2005，110，2： | 1 |
| 7 | A Novel Electronic Tongue Combined MLAPS with Stripping Voltammetry for Environment Detection 显示文摘 | Men Hong Zou Shaofang Li Yi | 2005 | Sensors and Actuators B2005，，110： | 1 |
| 8 | Bounds of divided universal Bernoulli numbers and universal Kummer congruences显示文摘 | Arnold Adelberg Shaofang Hong Wenli Ren | 2007 | Proceedings of the American Mathematical Society2007，，1： | 1 |
| 9 | New Lower Bounds for the Least Common Multiples of Arithmetic Progressions显示文摘For relatively prime positive integers u_0 and r,and for 0≤k≤n,define u_k:=u_0+kr.Let L_n:=1cm(u_0,u_1,…,u_n)and let a,l≥2 be any integers.In this paper,the authors show that,for integersα≥a,r≥max(a,l-1)and n≥lar,the following inequality holds L_n≥u_0r^((l-1)α+a-l)(r+1)~n.Particularly,letting l=3 yields an improvement on the best previous lower bound on L_n obtained by Hong and Kominers in 2010. | Rongjun WU Qianrong TAN Shaofang HONG | 2013 | Chinese Annals of Mathematics,Series B2013，34，6： | 1 |
| 10 | The p-adic approach to Wolstenholme's Theorem显示文摘 | SUN Qi HONG Shaofang | 2001 | Northeasterm Mathematical Journal2001，17，2： | 1 |
| 11 | A Curious Identity on Multiple Sums over Fields with Applications显示文摘Let F be a field,and let e,k be integers such that 1≤e≤|F\{0}|and k≥0.We show that for any subset{a1,……,ae}■F\{0},the curious identity∑(i1+……ie)∈Z^(e)≥0,i1+……+ie=k a_(1)^(i1)…a_(e)^(ie)=∑i=1 e a_(i)^(k+e-1)/∏i≠j=1 e(a_(i)-a_(j))holds with Z≥0 being the set of nonnegative integers.As an application,we prove that for any subset{a_(1)…,a_(e)}■F_(q)\{0}with F_(q)being the finite field of q elements and e,l being integers such that 2≤e≤q-1 and 0≤l≤e-2,∑(i_(1),…,i_(e))∈Z^(e)≥0,i_(1)+…i_(e)=q-e+l a_(1)^(i1)…a_(e)^(ie)=0 Using this identity and providing an extension of the principle of cross-classification that slightly generalizes the one obtained by Hong in 1996,we show that if r is an integer with 1≤r≤q-2,then for any subset{a_(1),…a_(r)}■F_(q)^(*)we have x^(q-1)-1/∏i=1 r(x-a_(i))-∑i=1 q-1-r(∑i_(1)+…+i_(r)=q-1-r-i^(a_(1)^(i1)…a_(r)^(ir)))x^(i).This implies#{x∈Fq*|∑i=0 q-1-4(∑_(i1)+…+ir=q-1-r-i^(a_(1)^(i1)…a_(r)^(ir)))x^(i)=0}=q-1-r. | Yongchao Xu Shaofang Hong | 2021 | Algebra Colloquium2021，28，2： | 1 |
| 12 | A novel electronic tongue combined MLAPS with stripping voltammetry for environment detection显示文摘 | Men Hong Zou Shaofang Li Yi | 2005 | Sensors and Actuators B2005，110，2： | 1 |
| 13 | A novel electronic tongue combined MLAP swith stripping vohammetry for environment detection显示文摘 | Men Hong Zou Shaofang Li Yi | 2005 | Sensors and Actuators B2005，110，2： | 1 |
| 14 | Automatic Monitoring Electronic Tongue with MEAs for Environmental Analysis显示文摘An automatic monitoring electronic tongue based on differential pulse stripping voltammetry (DPSV) was developed for heavy metals analysis.Simultaneous detections of trace Zn(Ⅱ),Cd(Ⅱ),Pb(Ⅱ),Cu(Ⅱ),Fe(Ⅲ) and Cr(Ⅲ) in water samples were performed with three electrochemical sensors.The sensor chip combined a silicon-based Hg-coated Au microelectrode array (MEA) as the working electrode on one side with an Ag/AgCl reference electrode and a Pt counter electrode on the other side.With a computer controlled multipotentiostat,pumps and valves,the electronic tongue realized in-situ real-time detection of the six metals mentioned above at parts-per-billion level without manual operation. | Shaofang Zou Hong Men Yi Li Yinping Wang Ping Wang | 2006 | 稀有金属材料与工程2006，35，A03： | 0 |
| 15 | A Note on the Tu-Deng Conjecture显示文摘Abstract Let k be a positive integer.For any positive integer x=∑∞i=0x i2i,where x i=0,1,we define the weight w(x)of x by w(x):=∑∞i=0x i.For any integer t with 0 | CHENG Kaimin HONG Shaofang ZHONG Yuanming | 2015 | Journal of Systems Science & Complexity2015，28，3： | 0 |
| 16 | On Ordinary Words of Standard Reed-Solomon Codes over Finite Fields显示文摘Reed-Solomon codes are widely used to establish a reliable channel to transmit information in digital communication which has a strong error correction capability and a variety of efficient decoding algorithm.Usually we use the maximum likelihood decoding(MLD)algorithm in the decoding process of Reed-Solomon codes.MLD algorithm relies on determining the error distance of received word.Dür,Guruswami,Wan,Li,Hong,Wu,Yue and Zhu et al.got some results on the error distance.For the Reed-Solomon code C,the received word u is called an ordinary word of C if the error distance d(u,C)=n-deg u(x)with u(x)being the Lagrange interpolation polynomial of u.We introduce a new method of studying the ordinary words.In fact,we make use of the result obtained by Y.C.Xu and S.F.Hong on the decomposition of certain polynomials over the finite field to determine all the ordinary words of the standard Reed-Solomon codes over the finite field of q elements.This completely answers an open problem raised by Li and Wan in[On the subset sum problem over finite fields,Finite Fields Appl.14(2008)911-929]. | Xiaofan Xu Shaofang Hong | 2021 | Algebra Colloquium2021，28，4： | 0 |
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