Optimal p-Ary Codes from One-Weight and Two-Weight Codes over IF_p+ vIF_p

This paper is devoted to determining the structures and properties of one-Lee weight codes and two-Lee weight projective codes Ck1,k2,k3 over p IF+ v IFp with type p2k1pk2pk3. The authors introduce a distance-preserving Gray map from( IFp + v IFp)nto2np. By the Gray map, the authors construct a family of optimal one-Hamming weight p-ary linear codes from one-Lee weight codes over IFp+ v IFp, which attain the Plotkin bound and the Griesmer bound. The authors also obtain a class of optimal p-ary linear codes from two-Lee weight projective codes over IFp + vIFp, which meet the Griesmer bound.     

The Determination on Weight Hierarchies of q-Ary Linear Codes of Dimension 5 in Class IV

The weight hierarchy of a linear[n;k;q]code C over GF(q) is the sequence(d_1,d_2,…,d_k)where d_r is the smallest support of any r-dimensional subcode of C. "Determining all possible weight hierarchies of general linear codes" is a basic theoretical issue and has important scientific significance in communication system.However,it is impossible for g-ary linear codes of dimension k when q and k are slightly larger,then a reasonable formulation of the problem is modified as: "Determine almost all weight hierarchies of general g-ary linear codes of dimension k".In this paper,based on the finite projective geometry method,the authors study g-ary linear codes of dimension 5 in class IV,and find new necessary conditions of their weight hierarchies,and classify their weight hierarchies into6 subclasses.The authors also develop and improve the method of the subspace set,thus determine almost all weight hierarchies of 5-dimensional linear codes in class IV.It opens the way to determine the weight hierarchies of the rest two of 5-dimensional codes(classes III and VI),and break through the difficulties.Furthermore,the new necessary conditions show that original necessary conditions of the weight hierarchies of k-dimensional codes were not enough(not most tight nor best),so,it is important to excogitate further new necessary conditions for attacking and solving the fc-dimensional problem.     

A MacWilliams type identity on Lee weight for linear codes over \mathbb{F}_2 + u\mathbb{F}_2^*

In this paper,the MacWilliams type identity for the m-ply Lee weight enumerator for linear codes over F₂ + uF₂ is determined.As an application of this identity,the authors obtain a MacWilliams type identity on Lee weight for linear codes over F_2^m+ uF_2^m.Furthermore,the authors prove a duality for the m-ply Lee weight distributions by taking advantage of the Krawtchouk polynomials.     

（1-uv）-CONSTACYCLIC CODES OVER Fp＋uFp＋vFp＋uvFp

Constacyclic codes are an important class of linear codes in coding theory.Many optimal linear codes are directly derived from constacyclic codes.In this paper,（1 — uv）-constacyclic codes over the local ring Fp ＋ uFp ＋ vFp ＋ uvFp are studied.It is proved that the image of a（1 — uv）-constacyclic code of length n over Fp ＋ uFp ＋ vFp ＋ uvFp under a Gray map is a distance invariant quasi-cyclic code of index p2 and length p3n over Fp.Several examples of optimal linear codes over Fp from（1 — uv）-constacyclic codes over Fp ＋ uFp ＋ vFp ＋ uvFp are given.     

Study on cipher propertys of constant weight codes

1 .INTRODUCTIONIn the research on constant weight codes , someconsiderable headway have been made in the do-main of the binary constant weight codes . Espe-cially ,people focus attention on the functionA(n,d, w) ,which denotes the maxi mumcardinality ofa binary code of lengthn,mini mumdistancedandconstant weightw.Some newresults on upper andlower bounds onA( n, d, w)are presented inRefs .[1 ~4] ,and the result on extensive constantweight codes is givenin Refs .[5 ~10] ,respective-ly .In …     

Optimal and suboptimal structured algorithms of binary linear block codes

The optimal and suboptimal structured algorithms of linear block codes from the geometrical perspective are represented.The minimum distance and weight property lemmas and the theorem are proved for the generator matrix.Based upon the property of generator matrix,the structured algorithms of linear block codes are demonstrated.Since the complexity of optimal structured algorithm is very high,the binary linear block codes is searched by using the suboptimal structured algorithm.The comparison with Bose-Chaudhuri-Hocquenqhem(BCH) codes shows that the searched linear block codes are equivalent on minimum distance and can be designed for more block lengths.Because the linear block codes are used widely in communication systems and digital applications,the optimal and suboptimal structured algorithms must have great future being widely used in many applications and perspectives.     

THE HAMMING DISTANCES OF NEGACYCLIC CODES OF LENGTH 2^s OVER GR （2^α, m）

The Hamming distances of all negacyclic codes of length 28 over the Gaiois ring GR （2^α, m） are given. In particular, the Lee distances of all negacyclic codes over Z4 of length 2^s are obtained. The Gray images of such negacyclic codes over Z4 are also determined under the Gray map.     

Good p-ary quasic-cyclic codes from cyclic codes over $$\mathbb{F}_ p + v\mathbb{F}_ p$$

This paper introduces a Gray map from（F_p + uF_p）ⁿ to F_p²ⁿ,and describes the relationship between codes over F_p + vF_p and their Gray images.The authors prove that every cyclic code of arbitrary length n over F_p + vF_p is principal,and determine its generator polynomial as well as the number of cyclic codes.Moreover,the authors obtain many best-known p-ary quasic-cyclic codes in terms of their parameters via the Gray map.     

The projective plane crossing numbers of circular graphs*

The authors give an upper bound for the projective plane crossing number of a circular graph. Also, the authors prove the projective plane crossing numbers of circular graph C （8, 3） and C （9, 3） are 2 and 1, respectively.     

A family of constacyclic codes over F 2 + uF 2 + vF 2 + uvF 2

This paper studies（1+u）-constacyclic codes over the ring F₂+uF₂+vF₂+uvF₂- It is proved that the image of a（1 + u）-constacyclic code of length n over F₂+uF₂+vF₂+uvF₂ under a Gray map is a distance invariant binary quasi-cyclic code of index 2 and length An.A set of generators of such constacyclic codes for an arbitrary length is determined.Some optimal binary codes are obtained directly from（1 + u）-constacyclic codes over F₂+uF₂+vF₂+uvF₂.     

On the relative generalized hamming weights of a 4-dimensional linear code and a subcode with dimension one

Finite projective geometry method is effectively used to study the relative generalized Hamming weights of 4-dimensional linear codes,which are divided into 9 classes in order to get much more information about the relative generalized Hamming weights,and part of the relative generalized Hamming weights of a 4-dimensional linear code with a 1-dimensional subcode are determined.     

On LCD Negacyclic Codes over Finite Fields

This paper deduces the structure of LCD negacyclic codes over the finite field Fq, where q is an odd prime power. Based on the study of q-cyclotomic cosets modulo 2 n, the authors obtain the parameters of LCD negacyclic codes of lengths n =(q+1)/2,(q~m-1)/2(q-1)and q~(t·2~τ)-1/2(q~t +1), respectively. And many optimal codes are given. Moreover, the authors research two special classes of MDS LCD negacyclic codes of length n |(q-1)/2 and n |(q+1)2, respectively.     

NNMDS codes

C is an[n,k,d]_q linear code over F₉.And s（C）=n+1-k-d is the Singleton defect of C.An MDS code C with s（C）=0 has been studied extensively.Recently,a near-MDS code C with s（C）=s（C^⊥）=1 is studied by many scholars,where C^⊥denotes the dual code of C.This paper concentrates on the linear code C with s（C）=s（C^⊥）=2,and the author calls it an NNMDS code.A series of iff conditions of NNMDS codes are presented.And the author gives an upper bound on length of NNMDS codes.In the last,some examples of NNMDS are given.     

ON THE SECOND GREEDY WEIGHT FOR LINEARCODES SATISFYING THE FULLRANK CONDITION

The maximum of g2-d2 for linear [n, k, d; q] codes C is studied. Here d2 is the smallest size of the support of 2-dimensional subcodes of C and g2 is the smallest size of the support of 2-dimensional subcodes of C which contains a codeword of weight d. The extra cost to the greedy adversary to get two symbols of information using some algorithm is g2-d2. For codes satisfying the fullrank condition of general dimensions, upper bounds on the maximum of g2-d2 are given. Under some condition we have got code C where g2-d2 reaches the upper bound.     

A Class of Constacyclic Codes over Ring R+vR

By constructing a Gray map,a class of constacyclic codes over ring=■+vR is studied.Using cyclic codes and negacyclic codes of length p~s over ring R,the structure of(1-2v)-constacyclic codes and dual codes of length p~s over ring are given,the Gray images of(1-2v)-constacyclic codes in a particular case are also studied.It is shown that linear codes of length p~s over ring ■ are(1-2v)-constacyclic codes if and only if their Gray images are distance-invariant cyclic codes of length2p~s over ring R.     

Inverse Center Location Problem on a Tree

This paper discusses the inverse center location problem restricted on a tree with different costs and bound constraints. The authors first show that the problem can be formulated as a series of combinatorial linear programs, then an O（｜V｜^2 log ｜V｜） time algorithm to solve the problem is presented. For the equal cost case, the authors further give an O（｜V｜） time algorithm.     

A note on two classes of Boolean functions with optimal algebraic immunity

Tu and Deng proposed a class of bent functions which are of optimal algebraic immunity under the assumption of a combinatorial conjecture.In this paper,the authors compute the dual of the Tu-Deng functions and then show that they are still of optimal algebraic immunity under the assumption of the same conjecture.For another class of Boolean functions constructed by Tang,et al.which are of optimal algebraic immunity with similar forms to Tu-Deng functions,the authors show that they are not bent functions by using some basic properties of binary complete Kloosterman sums.     

Strong n-discount and finite-horizon optimality for continuous-time Markov decision processes

This paper studies the strong n（n =-1,0）-discount and finite horizon criteria for continuoustime Markov decision processes in Polish spaces.The corresponding transition rates are allowed to be unbounded,and the reward rates may have neither upper nor lower bounds.Under mild conditions,the authors prove the existence of strong n（n =-1,0）-discount optimal stationary policies by developing two equivalence relations：One is between the standard expected average reward and strong-1-discount optimality,and the other is between the bias and strong 0-discount optimality.The authors also prove the existence of an optimal policy for a finite horizon control problem by developing an interesting characterization of a canonical triplet.     

Distribution of one-error linear complexity of binary sequences for arbitrary prime period

Complexity measures for sequences,such as the linear complexity and the k-error linear complexity,play an important role in stream ciphers.This contribution studies the distribution of 1-error linear complexity of binary sequences with arbitrary prime period.For any odd prime N,the authors present all possible values of 1-error linear complexity of N-periodic binary sequences,and derive the exact formulas to count the number of N-periodic binary sequences with any given 1-error linear complexity.