decoding cyclic codes up to a new bound on the minimum distance a new lower bound on the minimum distance of q ary cyclic codes is proposed this bound improves upon the bose chaudhuri hocquenghem bound and for some codes upon the hartmann tzeng bound several boston bounds are special cases of our bound for some classes of codes the bound on the minimum distance is refined furthermore a quadratic time decoding algorithm up to this new bound is developed the determination of the error locations is based on the euclidean algorithm and a modified chien search the error evaluation is done by solving a generalization of forney s formula

three classes of minimal linear codes over the finite fields of odd characteristic minimal linear codes are a special subclass of linear codes and have significant applications in secret sharing and secure two party computation in this paper we focus on constructing minimal linear codes with frac w min w max leq frac p 1 p for any odd prime p based on a generic construction of linear codes where w min and w max denote the minimum and maximum nonzero weights in a code respectively first we present two new infinite families of minimal linear codes with two or three weights by selecting suitable subcode of linear codes which are not minimal second we also present an infinite family of minimal linear codes by employing partial spreads which can be viewed as a generalization of the construction of ding et al in addition we determine the weight distributions of all these minimal linear codes

the linear complexity of new generalized cyclotomic binary sequences of order four whiteman generalized cyclotomic sequences are proven to exhibit a number of good randomness properties in this paper we determine the linear complexity of some newly generalized cyclotomic sequences of order four with period pq which are defined by ding and helleseth the results show that all of these sequences have high linear complexity

constacyclic codes cocycles and a u v u v construction a connection between cohomology cocycles and constacyclic codes is explored it suggests an isomorphism between cyclic codes of length mn and a direct sum of m constacyclic codes of length n the isomorphism is used i to study the discrete fourier transforms and the decomposition of group ring codes ii to give a u v u v construction over gf q when q is odd the u v u v construction gives some of the best ternary cyclic codes and classes of nearly mds cyclic codes of length 2q 2 the symmetry of the construction is used to give a new proof that there are no cyclic self dual codes over gf q when q is odd

cyclic self dual codes it is shown that if the automorphism group of a binary self dual code satisfies a certain condition then the code contains words of weight congruent to 2 modulo 4 in particular no cyclic binary self dual code can have all its weights divisible by four the number of cyclic binary self dual codes of length n is determined and the shortest nontrivial code in this class is shown to have length 14

on the algebraic structure of quasi cyclic codes i finite fields a new algebraic approach to quasi cyclic codes is introduced the key idea is to regard a quasi cyclic code over a field as a linear code over an auxiliary ring by the use of the chinese remainder theorem crt or of the discrete fourier transform dft that ring can be decomposed into a direct product of fields that ring decomposition in turn yields a code construction from codes of lower lengths which turns out to be in some cases the celebrated squaring and cubing constructions and in other cases the u spl upsi u spl upsi and vandermonde constructions all binary extended quadratic residue codes of length a multiple of three are shown to be attainable by the cubing construction quinting and septing constructions are introduced other results made possible by the ring decomposition are a characterization of self dual quasi cyclic codes and a trace representation that generalizes that of cyclic codes

secret sharing schemes from binary linear codes in principle every linear code can be used to construct a secret sharing scheme however in general determining the access structure of the scheme is very hard on the other hand finding error correcting codes that produce secret sharing schemes with efficient access structures is also difficult in this paper we study a set of minimal codewords for certain classes of binary linear codes and then determine the access structure of secret sharing schemes based on these codes furthermore we prove that the secret sharing schemes obtained are democratic in the sense that every participant is involved in the same number of minimal access sets

linear codes from some 2 designs a classical method of constructing a linear code over mathrm gf q with a t design is to use the incidence matrix of the t design as a generator matrix over mathrm gf q of the code this approach has been extensively investigated in the literature in this paper a different method of constructing linear codes using specific classes of 2 designs is studied and linear codes with a few weights are obtained from almost difference sets difference sets and a type of 2 designs associated to semibent functions two families of the codes obtained in this paper are optimal the linear codes presented in this paper have applications in secret sharing and authentication schemes in addition to their applications in consumer electronics communication and data storage systems a coding theory approach to the characterization of highly nonlinear boolean functions is presented

duadic codes a new family of binary cyclic n n 1 2 and n n 1 2 codes are introduced which include quadratic residue qr codes when n is prime these codes are defined in terms of their idempotent generators and they exist for all odd n p 1 a 1 p 2 a 2 cdots p r a r where each p i is a prime equiv pm 1 pmod 8 dual codes are identified the minimum odd weight of a duadic n n 1 2 code satisfies a square root bound when equality holds in the sharper form of this bound vectors of minimum weight hold a projective plane the unique projective plane of order 8 is held by the minimum weight vectors in two inequivalent 73 37 9 duadic codes all duadic codes of length less than 127 are identified and the minimum weights of their extensions are given one of the duadic codes of length 113 has greater minimum weight than the qr code of that length

the weight distribution of some irreducible cyclic codes irreducible cyclic codes have been an interesting subject of study for a long time their weight distribution is known in only a few cases in this paper the weight distribution of the irreducible cyclic codes in a number of other cases is determined the number of nonzero weights in the codes dealt with in this paper varies between one and four

the weight distribution of a class of linear codes from perfect nonlinear functions in this correspondence the weight distribution of a class of linear codes based on perfect nonlinear functions also called planar functions is determined the class of linear codes under study are either optimal or among the best codes known and have nice applications in cryptography

cryptographic properties of some binary generalized cyclotomic sequences with the length p2 herein the cryptographic properties of some generalized cyclotomic sequences with the period p 2 are investigated results show that these sequences always possess good balance and linear complexity larger than half the period their autocorrelation functions are shown to be three valued if p 3 mod4 and five valued otherwise moreover the distribution of patterns of length 2 are discussed

on a new homotopy continuation trajectory for nonlinear complementarity problems most known continuation methods for p 0 complementarity problems require some restrictive assumptions such as the strictly feasible condition and the properness condition to guarantee the existence and the boundedness of certain homotopy continuation trajectory to relax such restrictions we propose a new homotopy formulation for the complementarity problem based on which a new homotopy continuation trajectory is generated for p 0 complementarity problems the most promising feature of this trajectory is the assurance of the existence and the boundedness of the trajectory under a condition that is strictly weaker than the standard ones used widely in the literature of continuation methods particularly the often assumed strictly feasible condition is not required here when applied to p complementarity problems the boundedness of the proposed trajectory turns out to be equivalent to the solvability of the problem and the entire trajectory converges to the unique least element solution provided that it exists moreover for monotone complementarity problems the whole trajectory always converges to a least 2 norm solution provided that the solution set of the problem is nonempty the results presented in this paper can serve as a theoretical basis for constructing a new path following algorithm for solving complementarity problems even for the situations where the solution set is unbounded

a new construction of frequency hopping sequences with optimal partial hamming correlation frequency hopping sequences fhss with favorable partial hamming correlation properties have important applications in many synchronization and multiple access systems in this paper lower bounds on the partial hamming correlation of fhss and fhs sets are proposed they slightly improve the known bounds by eun et al and zhou et al a construction of fhss and fhs sets having optimal partial hamming correlation with respect to the improved bounds is also presented based on the theory of generalized cyclotomy our construction yields optimal fhss and fhs sets with new and flexible parameters not covered in this paper

strictly optimal frequency hopping sequence sets with optimal family sizes frequency hopping sequences fhss with favorable partial hamming correlation properties are desirable in many synchronization and multiple access systems an fhs set is said to be strictly optimal if it has optimal partial hamming correlation for any correlation window in this paper we derive upper bounds on the family sizes of fhs sets with respect to partial hamming correlation from some classical bounds on error correcting codes we then present strictly optimal fhs sets having optimal family sizes with respect to one of the new bounds in particular our construction gives new parameters not covered in the literature

a new bound for the minimum distance of a cyclic code from its defining set a new lower bound for the distance of cyclic codes is proposed this bound depends on the defining set of the code like several other bounds the proposed bound improves upon the bose chaudhuri hocquehghen bch bound and for some codes improves upon the hartmann tzeng bound and the roos bound as well

a linear code based on resilient boolean maps whose dual is a platform for a robust secret sharing scheme abstract a linear code based on resilient maps on vector spaces over finite fields is introduced and lower bounds for its minimal distance are given the dual code is used to define a secret sharing scheme sss according to massey construction since the dual code satisfies the conditions stated by padro and saez the generalization to a vector space sss is robust against cheaters

zero difference balanced functions with new parameters and their applications as an optimal combinatorial object zero difference balanced zdb functions introduced by ding in 2008 are a generalization of the well known perfect nonlinear functions zdb functions have received much attention in recent years due to its important applications in coding theory and sequence design one objective of this paper is to present a construction of zdb functions based on a kind of generalized cyclotomy it generates zdb functions over cyclic group with new parameters which can not be produced by earlier constructions another objective of this paper is to employ these zdb functions to obtain at the same time 1 optimal constant composition codes 2 perfect difference systems of sets and 3 optimal frequency hopping sequences all with new parameters

a unified approach to whiteman s and ding helleseth s generalized cyclotomy over residue class rings the theory of cyclotomy dates back to gauss and has a number of applications in combinatorics coding theory and cryptography cyclotomy over a residue class ring bbzv can be divided into classical cyclotomy or generalized cyclotomy depending on v prime or composite in this paper we introduce a generalized cyclotomy of order d over bbzp1e1p2e2 pnen which includes whiteman s and ding helleseth s generalized cyclotomy as special cases here p1 p2 pn are pairwise distinct odd primes satisfying d pi 1 for all 1 i n and e1 e2 en are positive integers we derive some basic properties of the corresponding cyclotomic numbers and obtain a general formula to compute them via classical cyclotomic numbers as applications we completely solve an open problem and a conjecture on whiteman s generalized cyclotomy of order four over bbzp1p2 besides we also construct an infinite series of near optimal codebooks over bbzp1p2 as well as some infinite series of asymptotically optimal difference systems of sets over bbzp1e1p2e2 pnen

five families of three weight ternary cyclic codes and their duals as a subclass of linear codes cyclic codes have applications in consumer electronics data storage systems and communication systems as they have efficient encoding and decoding algorithms in this paper five families of three weight ternary cyclic codes whose duals have two zeros are presented the weight distributions of the five families of cyclic codes are settled the duals of two families of the cyclic codes are optimal

on the climatic dependence of the iuh a rainfall runoff analysis of the nash model and the geomorphoclimatic theory the geomorphoclimatic theory of the iuh is compared with the results obtained through derivation of the iuh s from a series of rainfall runoff experiments conducted via simulation for one basin and three different climates the theoretical distributions of the peak and time to peak of the iuh compare very well with those obtained in the experiments the peak and time to peak of the response hydrograph estimated through simple formulas which assume a triangular iuh also have a good agreement with the values obtained through the rainfall runoff model it is shown that the uncertainty due to the variability of the iuh with the rainfall input may produce a sizable variance of estimation in the outflow hydrograph when working under the assumed nonlinear framework the analysis is carried out with constant roughness and rectangular channels characteristics which are taken independent of the size of the discharge

the minimum distance of the duals of binary irreducible cyclic codes irreducible cyclic codes have been an interesting subject of study for many years the weight distribution of some of them have been determined we determine the minimum distance and certain weights of the duals of binary irreducible cyclic codes we show that the weight distribution of these codes is determined by the cyclotomic numbers of certain order as a byproduct we describe a class of double error correcting codes

a family of five weight cyclic codes and their weight enumerators cyclic codes are a subclass of linear codes and have applications in consumer electronics data storage systems and communication systems as they have efficient encoding and decoding algorithms in this paper a family of p ary cyclic codes whose duals have three pairwise nonconjugate zeros is proposed the weight distribution of this family of cyclic codes is determined it turns out that the proposed cyclic codes have five nonzero weights

a new lower bound for the minimum distance of a cyclic code a new lower bound for the minimum distance of a linear code is derived when applied to cyclic codes both the bose chaudhuri hocquenghem bch bound and the hartmann tzeng ht bound are obtained as corollaries examples for which the new bound is superior to these two bounds as well as to the carlitz uchiyama bound are given

on the covering structures of two classes of linear codes from perfect nonlinear functions in this paper the weight distributions of two classes of linear codes based on all known explicit perfect nonlinear functions from fqm to itself are determined using a unified approach all the minimal codewords of these codes are characterized according to their weights which suggests that their covering structures are determined finally all the minimal access sets of the secret sharing schemes based on their dual codes are obtained

on the minimum distance of ternary cyclic codes there are many ways to find lower bounds for the minimum distance of a cyclic code based on investigation of the defining set some new theorems are derived these and earlier techniques are applied to find lower bounds for the minimum distance of ternary cyclic codes furthermore the exact minimum distance of ternary cyclic codes of length less than 40 is computed numerically a table is given containing all ternary cyclic codes of length less than 40 and having a minimum distance exceeding the bch bound it seems that almost all lower bounds are equal to the minimum distance especially shifting which is also done by computer seems to be very powerful for length 40

algebraic geometry codes the theory of error correcting codes derived from curves in an algebraic geometry was initiated by the work of goppa as generalizations of bose chaudhuri hocquenghem bch reed solomon rs and goppa codes the development of the theory has received intense consideration since that time and the purpose of the paper is to review this work elements of the theory of algebraic curves at a level sufficient to understand the code constructions and decoding algorithms are introduced code constructions from particular classes of curves including the klein quartic elliptic and hyperelliptic curves and hermitian curves are presented decoding algorithms for these classes of codes and others are considered the construction of classes of asymptotically good codes using modular curves is also discussed

on the minimum distance of cyclic codes the main result is a new lower bound for the minimum distance of cyclic codes that includes earlier bounds i e bch bound ht bound roos bound this bound is related to a second method for bounding the minimum distance of a cyclic code which we call shifting this method can be even stronger than the first one for all binary cyclic codes of length with two exceptions we show that our methods yield the true minimum distance the two exceptions at the end of our list are a code and its even weight subcode we treat several examples of cyclic codes of length geq 63

hermitian self dual abelian codes hermitian self dual abelian codes in a group ring fq2 g where fq2 is a finite field of order q2 and g is a finite abelian group are studied using the well known discrete fourier transform decomposition for a semisimple group ring a characterization of hermitian self dual abelian codes in fq2 g is given together with an alternative proof of necessary and sufficient conditions for the existence of such a code in fq2 g i e there exists a hermitian self dual abelian code in fq2 g if and only if the order of g is even and q 2l for some positive integer l later on the study is further restricted to the case where f22l g is a principal ideal group ring or equivalently g a z2k with 2 a based on the characterization obtained the number of hermitian self dual abelian codes in f22l a z2k can be determined easily when a is cyclic this result answers an open problem of jia et al concerning hermitian self dual cyclic codes in many cases f22l a z2k contains a unique hermitian self dual abelian code the criteria for such cases are determined in terms of l and the order of a finally the distribution of finite abelian groups a such that a unique hermitian self dual abelian code exists in f22l a z2 is established together with the distribution of odd integers m such that a unique hermitian self dual cyclic code of length 2 m over f22l exists

the linear complexity of whiteman s generalized cyclotomic sequences of period p m 1 q n 1 in this paper we mainly get three results first let p q be distinct primes with p 1 p q 1 q p 1 q 1 e we give a method to compute the linear complexity of whiteman s generalized cyclotomic sequences of period pm 1qn 1 second if e 4 we compute the exact linear complexity of whiteman s generalized cyclotomic sequences third if p q 5 mod 8 gcd p 1 q 1 4 and we fix a common primitive root g of both p and q then 2 h0 g which is a subgroup of the multiplicative group z pq if and only if whiteman s generalized cyclotomic numbers of order 4 depend on the decomposition pq a2 4b2 with 4 b

a geomorphoclimatic theory of the instantaneous unit hydrograph the instantaneous unit hydrograph is conceived as a random function of climate and geomorphology varying with the characteristics of the rainfall excess the probability density functions of the peak and time to peak of the iuh are analytically derived as functions of the rainfall characteristics and the basin geomorphological parameters the main characteristics of these pdf s are studied and a new approach to hydrologic similarity is initiated under the concept of the geomorphoclimatic iuh for a given set of geomorphologic characteristics and a particular intensity and duration of rainfall the peak and time to peak of the iuh corresponding to those values can be easily estimated

a commutative algebra approach to linear codes abstract recently some methods have been proposed to find the distance and weight distribution of cyclic codes using grobner bases we identify a class of codes for which these methods can be generalized we show that this class contains all interesting linear codes and we provide variants and improvements this approach sometimes reveals an unexpected algebraic structure in the code we also investigate the decoding for a subclass proving the existence of general error locator polynomials

on self dual cyclic codes over finite fields in coding theory self dual codes and cyclic codes are important classes of codes which have been extensively studied the main objects of study in this paper are self dual cyclic codes over finite fields i e the intersection of these two classes we show that self dual cyclic codes of length n over bbfq exist if and only if n is even and q 2m with m a positive integer the enumeration of such codes is also investigated when n and q are even there is always a trivial self dual cyclic code with generator polynomial xn 2 1 we therefore classify the existence of self dual cyclic codes for given n and q into two cases when only the trivial one exists and when two or more such codes exist given n and m an easy criterion to determine which of these two cases occurs is given in terms of the prime factors of n for most n we also show that over a fixed field the latter case occurs more frequently as the length grows

narrow sense bch codes over mathrm gf q with length n frac q m 1 q 1 cyclic codes are widely employed in communication systems storage devices and consumer electronics as they have efficient encoding and decoding algorithms bch codes as a special subclass of cyclic codes are in most cases among the best cyclic codes a subclass of good bch codes are the narrow sense bch codes over mathrm gf q with length n q m 1 q 1 little is known about this class of bch codes when q 2 the objective of this paper is to study some of the codes within this class in particular the dimension the minimum distance and the weight distribution of some ternary bch codes with length n 3 m 1 2 are determined in this paper a class of ternary bch codes meeting the griesmer bound is identified an application of some of the bch codes in secret sharing is also investigated

properties and applications of preimage distributions of perfect nonlinear functions the preimage distributions of perfect nonlinear functions from an abelian group of order n to an abelian group of order 3 or 4 respectively are studied based on the properties of the preimage distributions of perfect nonlinear functions from an abelian group of order 3r to an abelian group of order 3 the weight distributions of the ternary linear codes c pi from the perfect nonlinear functions pi x from f 3 r to itself are determined these results suggest that two open problems proposed by carlet ding and yuan in 2005 and 2006 respectively are answered

a construction of multiple optimal zcz sequence sets with good cross correlation zero correlation zone zcz sequences are a class of spreading sequences having ideal auto correlation and cross correlation in a zone around the origin they have been extensively studied in recent years due to their important applications in quasi synchronous code division multiple access systems in this paper a construction of zcz sequence sets is proposed based on perfect nonlinear functions it generates multiple zcz sequence sets with the properties 1 each sequence is perfect in the sense that its out of phase auto correlation is always zero 2 each zcz sequence set is optimal with respect to the tang fan matsufuji bound in which all the sequences are pairwise cyclically distinct and 3 the maximum inter set cross correlation of multiple sequence sets achieves the well known sarwate bound

linear complexity of binary whiteman generalized cyclotomic sequences of order 2k in this correspondence we obtain the linear complexity and minimal polynomials of binary whiteman generalized cyclotomic sequences of order 2 k where k 1 our result shows that all of these sequences possess large linear complexity

abelian codes in principal ideal group algebras we study abelian codes in principal ideal group algebras pigas we first give an algebraic characterization of abelian codes in any group algebra and provide some general results for abelian codes in a piga which can be viewed as cyclic codes over a semisimple group algebra it is shown that every abelian code in a piga admits generator and check elements these are analogous to the generator and parity check polynomials of cyclic codes a characterization and an enumeration of euclidean self dual and euclidean self orthogonal abelian codes in a piga are given which generalize recent analogous results for self dual cyclic codes in addition the structures of reversible and complementary dual abelian codes in a piga are established again extending results on reversible and complementary dual cyclic codes finally asymptotic properties of abelian codes in a piga are studied an upper bound for the minimum distance of abelian codes in a non semisimple piga is given in terms of the minimum distance of abelian codes in semisimple group algebras abelian codes in a non semisimple piga are then shown to be asymptotically bad similar to the case of repeated root cyclic codes

two recursive algorithms for computing the weight distribution of certain irreducible cyclic codes two recursive algorithms for computing the weight distributions of certain binary irreducible cyclic codes of length n in the so called index 2 case are presented the running times of these algorithms are smaller than o log sup 2 r where r 2 sup m and n is a factor of r 1

the covering radius of cyclic codes of length up to 31 corresp the covering radius is given for all binary cyclic codes of length less than or equal to 31 many of these codes are optimal in the sense of having the smallest possible covering radius of any linear code of that length and dimension

spectral residual methods with two new non monotone line searches for large scale nonlinear systems of equations in this study we develop two new non monotone line searches and combine them into a spectral residual method furthermore we apply these methods to complementary problems and discuss global convergence under several suitable conditions numerical experiments support the efficiency of the presented methods

value distributions of exponential sums from perfect nonlinear functions and their applications in this paper we present a unified way to determine the values and their multiplicities of the exponential sums sigmaxisinf q zetap tr af x bx a bisinfq q pm pges3 for all perfect nonlinear functions f which is a dembowski ostrom polynomial or p 3 f x 3 k 1 2 where k is odd and k m 1 as applications we determine 1 the correlation distribution of the m sequence alambda tr gammalambda lambda 0 1 and the sequence blambda tr f gammalambda lambda 0 1 over fp where gamma is a primitive element of fq and 2 the weight distributions of the linear codes over fp defined by f

variational convergence of bifunctions motivating applications it is shown that a number of variational and equilibrium problems can be cast as finding the maxinf points or minsup points of bivariate functions for short bifunctions these problems include linear and nonlinear complementarity problems fixed points variational inequalities inclusions noncooperative games and walras and nash equilibrium problems one appeals to the theory of lopsided convergence for bifunctions to derive stability results for each one of these problems

a new approach to constructing quadratic pseudo planar functions over mathbb f _ 2 n planar functions over finite fields give rise to finite projective planes they were also used in the constructions of des like iterated ciphers error correcting codes and codebooks they were originally defined only in finite fields with odd characteristic but recently zhou introduced pesudo planar functions in even characteristic which yields similar applications all known pesudo planar functions are quadratic and hence they give presemifields in this paper a new approach to constructing quadratic pseudo planar functions is given then five explicit families of pseudo planar functions are constructed one of which is a binomial two of which are trinomials and the other two are quadrinomials all known pesudo planar functions are revisited some of which are generalized these functions not only lead to projective planes relative difference sets and presemifields but also give optimal codebooks meeting the levenstein bound complete sets of mutually unbiased bases and compressed sensing matrices with low coherence

cyclic codes from the two prime sequences cyclic codes are a subclass of linear codes and have wide applications in consumer electronics data storage systems and communication systems as they have efficient encoding and decoding algorithms in this paper the two prime sequence is employed to construct several classes of cyclic codes over gf q lower bounds on the minimum weight of these cyclic codes are developed some of the codes obtained are optimal or almost optimal the p ranks of the twin prime difference sets and a class of almost difference sets are computed

on the relationships between perfect nonlinear functions and universal hash families in this paper the relationships between perfect nonlinear in brief pn functions and optimal universal hash families are discussed we point out the equivalence of constructions between them i e from pn functions one can obtain optimal universal hash families and vice versa as an application of our construction a message authentication code is proposed which provides better resistance to substitution attack than a known construction given by carlet et al in 2006 more generally the connections between functions with given differential uniformity and some universal hash families are studied

sharp p divisibility of weights in abelian codes over bbz p d bbz a theorem of mceliece on the p divisibility of hamming weights in cyclic codes over fp is generalized to abelian codes over zopf p dzopf this work improves upon results of helleseth kumar moreno shanbhag calderbank li poonen wilson and katz these previous attempts are not sharp in general i e do not report the full extent of the p divisibility except in special cases nor do they give accounts of the precise circumstances under which they do provide best possible results this paper provides sharp results on p divisibilities of hamming weights and counts of any particular symbol for an arbitrary abelian code over zopf p dzopf it also presents sharp results on 2 divisibilities of lee and euclidean weights for abelian codes over zopf 4zopf

seven classes of three weight cyclic codes cyclic codes are a subclass of linear codes and have applications in consumer electronics data storage systems and communication systems as they have efficient encoding and decoding algorithms compared with linear block codes in this paper seven classes of three weight cyclic codes over gf p whose duals have two zeros are presented where p is an odd prime the weight distributions of the seven classes of cyclic codes are settled some of the cyclic codes are optimal in the sense that they meet certain bounds on linear codes the application of these cyclic codes in secret sharing is also considered

differential spectrum of kasami power permutations over odd characteristic finite fields functions with low differential uniformity have important applications in cryptography coding theory and sequence design the differential spectrum of a cryptographic function is of great interest for estimating its resistance to some variants of differential cryptanalysis finding power permutations i e monomial bijective mappings over finite fields with low differential uniformity and determining their differential spectra have received a lot of attention over the past two decades the objective of this paper is to study the differential properties of the well known kasami power permutations x p 2k p k 1 over mathrm gf p n where p is an odd prime and k is an integer with gcd n k 1 it turns out that this family of monomials is differentially p 1 uniform our result in the case of p 3 gives an affirmative solution to a recent conjecture by xu cao and xu most notably the differential spectrum of this family of power permutations is completely determined

extension of the bch decoding algorithm to decode binary cyclic codes up to their maximum error correction capacities the bch algorithm is extended to correct more errors than indicated by the bch bound in the first step of the decoding procedure a number of errors corresponding to a particular case of the hartmann tzeng bound are corrected in the second step full error correction is the goal a measure for the worst case number of field elements of an extension field gf 2 sup m that must be tested for this purpose is given for binary cyclic linear unequal error protection codes as well as for conventional binary cyclic codes

k fold cyclotomy and its application to frequency hopping sequences for an integer k 1 let 1 qi k be prime powers such that qi mif 1 for some integers mi and f in this paper the k fold cyclotomy of fqk fqk as a nontrivial generalization of the conventional cyclotomy k 1 case and its application to frequency hopping sequences fhss are presented where fq is the finite field with q elements first the definitions of k fold cyclotomic classes and k fold cyclotomic numbers are given and then their basic properties including k fold diagonal sums are derived based on them new optimal fhs sets of length n and frequency set size m or m 1 with respect to the peng fan bound are constructed for a product n of distinct odd primes and a di visor m of n 1 furthermore new optimal fhss of length n and frequency set size m with respect to the lempel greenberger bound are constructed when n has at least one prime factor which is 3 modulo 4 and n 1 m is an even integer our constructions give several new optimal parameters not covered in the literature which are summarized in table i

generalized cyclotomic codes of length p sub 1 sup e 1 p sub t sup e t we first introduce a generalized cyclotomy of order 2 with respect to p sub 1 sup e 1 p sub t sup e t we then present two classes of new error correcting binary cyclic codes of length p sub 1 sup e 1 p sub t sup e t based on this generalized cyclotomy we prove either a square root bound or a similar bound on the minimum odd weight of those codes with length n p sub 1 sup e 1 p sub 2 sup e 2

linear codes with two or three weights from weakly regular bent functions linear codes with a few weights have applications in consumer electronics communication data storage system secret sharing authentication codes association schemes and strongly regular graphs this paper first generalizes the method of constructing two weight and three weight linear codes of ding et al and zhou et al to general weakly regular bent functions and determines the weight distributions of these linear codes it solves an open problem proposed by ding et al furthermore this paper constructs new linear codes with two or three weights and presents their weight distributions they contain some optimal codes meeting certain bound on linear codes

smoothing type algorithm for solving linear programs by using an augmented complementarity problem we present a smoothing type algorithm for solving the linear program lp by making use of an augmented system of its optimality conditions the algorithm is shown to be globally convergent without requiring any assumption it only needs to solve one system of linear equations and to perform one line search at each iteration in particular if the lp has a solution and hence it has a strictly complementary solution then the algorithm will generate a strictly complementary solution of the lp and if the lp is infeasible then the algorithm will correctly detect infeasibility of the lp to the best of our knowledge this is the first smoothing type algorithm for solving the lp having the above desired convergence features

optimal frequency hopping sequences of odd length in this paper a new generalized cyclotomy with respect to a positive odd integer is introduced and a construction of frequency hopping sequence sets and two constructions of frequency hopping sequences are proposed as its applications the frequency hopping sequence sets and frequency hopping sequences obtained in this paper can be optimal with respect to the peng fan bound and lempel greenberger bound respectively further the length of sequences in the optimal frequency hopping sequence sets can be any odd integer larger than 3 some of them have new parameters

proof of a conjecture on the sequence of exceptional numbers classifying cyclic codes and apn functions abstract we prove a conjecture that classifies exceptional numbers this conjecture arises in two different ways from cryptography and from coding theory an odd integer t 3 is said to be exceptional if f x x t is apn almost perfect nonlinear over f 2 n for infinitely many values of n equivalently t is exceptional if the binary cyclic code of length 2 n 1 with two zeros t has minimum distance 5 for infinitely many values of n the conjecture we prove states that every exceptional number has the form 2 i 1 or 4 i 2 i 1

hamming weights of the duals of cyclic codes with two zeros cyclic codes are an interesting type of linear codes and have wide applications in communication and storage systems due to their efficient encoding and decoding algorithms in this paper let bbb f r be a finite field with r q m suppose that g 1 g 2 in bbb f r are not conjugates over bbb f q operatorname ord g 1 n 1 operatorname ord g 2 n 2 d gcd n 1 n 2 and n n 1n 2 d let bbb f q g 1 bbb f q m 1 bbb f q g 2 bbb f q m 2 and text t i denote the trace function from bbb f q m i to bbb f q for i 1 2 we define a cyclic code mathcal c q m n 1 n 2 c a b a in bbb f q m 1 b in bbb f q m 2 where c a b text t 1 ag 1 0 text t 2 bg 2 0 text t 1 ag 1 1 text t 2 bg 2 1 ldots text t 1 ag 1 n 1 text t 2 bg 2 n 1 we mainly use gauss periods to present the weight distribution of the cyclic code mathcal c q m n 1 n 2 as applications we determine the weight distribution of cyclic code mathcal c q m q m 1 1 q m 2 1 with gcd m 1 m 2 1 in particular it is a three weight cyclic code if gcd q 1 m 1 m 2 1 we also explicitly determine the weight distributions of some classes of cyclic codes including several classes of four weight cyclic codes

optimal ternary cyclic codes from monomials cyclic codes are a subclass of linear codes and have applications in consumer electronics data storage systems and communication systems as they have efficient encoding and decoding algorithms perfect nonlinear monomials were employed to construct optimal ternary cyclic codes with parameters 3m 1 3m 1 2m 4 by carlet ding and yuan in 2005 in this paper almost perfect nonlinear monomials and a number of other monomials over gf 3m are used to construct optimal ternary cyclic codes with the same parameters nine open problems on such codes are also presented

linear codes from perfect nonlinear mappings and their secret sharing schemes in this paper error correcting codes from perfect nonlinear mappings are constructed and then employed to construct secret sharing schemes the error correcting codes obtained in this paper are very good in general and many of them are optimal or almost optimal the secret sharing schemes obtained in this paper have two types of access structures the first type is democratic in the sense that every participant is involved in the same number of minimal access sets in the second type of access structures there are a few dictators who are in every minimal access set while each of the remaining participants is in the same number of minimal access sets

on smoothing methods for the p 0 matrix linear complementarity problem in this paper we propose a big gamma smoothing method for solving the p0 matrix linear complementarity problem we study the trajectory defined by the augmented smoothing equations and global convergence of the method under an assumption that the original p0 matrix linear complementarity problem has a solution the method has been tested on the p0 matrix linear complementarity problem with unbounded solution set preliminary numerical results indicate the robustness of the method

constructions of optimal 2 d optical orthogonal codes via generalized cyclotomic classes optical orthogonal codes oocs are widely used as spreading codes in optical fiber networks in this paper a bound on the code size of 2 d oocs with both at most one pulse per wavelength am oppw and at most one pulse per time slot am oppts is derived accordingly two constructions of optimal 2 d oocs with both am oppw and am oppts are proposed via the generalized cyclotomic classes furthermore optimal 2 d ooc with am oppw can be also constructed by adding more codewords into the 2 d oocs with both am oppw and am oppts

from ds bounds for cyclic codes to true minimum distance for abelian codes in this paper we develop a technique to extend any bound for the minimum distance of cyclic codes constructed from its defining sets ds bounds to abelian or multivariate codes through the notion of mathbb b apparent distance we also study conditions for an abelian code to verify that its mathbb b apparent distance reaches its true minimum distance then we construct some codes as an application

on cyclic codes of composite length and the minimum distance in an interesting paper prof c ding provided three constructions of cyclic codes of length being a product of two primes numerical data shows that many codes from these constructions are best cyclic codes of the same length and dimension over the same finite field however not much is known about these codes in this paper we explain some of the numerical data by developing a general method on cyclic codes of composite length and on estimating the minimum distance we also provide a general construction of cyclic codes of composite length which are related to ding s constructions numerical data shows that it produces many best cyclic codes as well finally we point out how these cyclic codes can be used to construct convolutional codes with large free distance

cyclotomic constructions of cyclic codes with length being the product of two primes cyclic codes are an interesting type of linear codes and have applications in communication and storage systems due to their efficient encoding and decoding algorithms in this paper three types of generalized cyclotomy of order two are described and three classes of cyclic codes of length n1 n2 and dimension n1 n2 1 2 are presented and analyzed where n1 and n2 are two distinct primes bounds on their minimum odd like weight are also proved some of the codes presented in this paper are among the best cyclic codes

low degree planar polynomials over finite fields of characteristic two abstract planar functions are mappings from a finite field f q to itself with an extremal differential property such functions give rise to finite projective planes and other combinatorial objects there is a subtle difference between the definitions of these functions depending on the parity of q and we consider the case that q is even we classify polynomials of degree at most q 1 4 that induce planar functions on f q by showing that such polynomials are precisely those in which the degree of every monomial is a power of two as a corollary we obtain a complete classification of exceptional planar polynomials namely polynomials over f q that induce planar functions on infinitely many extensions of f q the proof strategy is to study the number of f q rational points of an algebraic curve attached to a putative planar function our methods also give a simple proof of a new partial result for the classification of almost perfect nonlinear functions