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+...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.展开更多
Abstract For relatively prime positive integers u0 and r, and for 0 〈 k ≤ n, define uk ：= u0 ＋ kr. Let Ln ：= 1cm（u0,u1,... ,un） and let a,l≥2 be any integers. In this paper, the authors show that, for integers...Abstract For relatively prime positive integers u0 and r, and for 0 〈 k ≤ n, define uk ：= u0 ＋ kr. Let Ln ：= 1cm（u0,u1,... ,un） and let a,l≥2 be any integers. In this paper, the authors show that, for integers α≥ a, r ≥max（a,l - 1） and n ≥lατ, the following inequality holds Ln≥u0r^（l-1）α＋a-l（r＋1）^n.Particularly, letting l = 3 yields an improvement on the best previous lower bound on Ln obtained by Hong and Kominers in 2010.展开更多
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 ...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].展开更多
基金supported partially by the National Science Foundation of China(Grant#11771304)the Fundamental Research Funds for the Central Universities.
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.
基金supported by the National Natural Science Foundation of China(No.10971145)the Ph.D.Programs Foundation of Ministry of Education of China(No.20100181110073)the Science&Technology Program of Sichuan Province(No.2013JY0125)
文摘Abstract For relatively prime positive integers u0 and r, and for 0 〈 k ≤ n, define uk ：= u0 ＋ kr. Let Ln ：= 1cm（u0,u1,... ,un） and let a,l≥2 be any integers. In this paper, the authors show that, for integers α≥ a, r ≥max（a,l - 1） and n ≥lατ, the following inequality holds Ln≥u0r^（l-1）α＋a-l（r＋1）^n.Particularly, letting l = 3 yields an improvement on the best previous lower bound on Ln obtained by Hong and Kominers in 2010.
基金supported by the National Science Foundation of China Grant 11771304Fundamental Research Funds for the Central Universities.X.F.Xu was partially supported by Foundation of Sichuan Tourism University Grant 20SCTUTY01.
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].
