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We prove that the covering radius of the Melas code $M(m,q)$ of length $n=q^{m}-1$ over $\mathbb {F}_{q}$ is 2 if $q > 3$. We also prove that the covering radius of $M(m,3)$ is 3 is $m \ge 3$ , the covering radius of $M(2,3)$ is 4, and the covering radii of $M(1,2)$ and $M(1,3)$ are 1. [ABSTRACT FROM AUTHOR]

Sequences and their correlation properties have been extensively studied due to their broad applications. In this paper, we develop a connection between sequences and well-studied combinatorial objects, circular Florentine arrays. This connection allows us to derive two types of sequences with good correlation properties. The first type consists of sequences having optimal correlation with respect to the Sarwate bound. Our constructions are based on perfect polyphase sequences. The number of perfect sequences with optimal correlation depends on the existence of circular Florentine arrays, which improves the previous known results. The second type is about multiple ZCZ sequence sets with low inter-set cross-correlation. Each generated ZCZ sequence set is optimal with respect to the Tang-Fan-Matsufuji bound, and each sequence in each set is perfect. In addition, any two sequences from distinct ZCZ sequence sets possess optimal inter-set cross-correlation with respect to the Sarwate bound. Compared with the previous results, the number of ZCZ sequence sets with optimal inter-set cross-correlation property is improved, because of the existence of circular Florentine arrays. [ABSTRACT FROM AUTHOR]

A new method is used to resolve a long-standing conjecture of Niho concerning the crosscorrelation spectrum of a pair of maximum length linear recursive sequences of length $2^{2m}-1$ with relative decimation $d=2^{m+2}-3$ , where $m$ is even. The result indicates that there are at most five distinct crosscorrelation values. Equivalently, the result indicates that there are at most five distinct values in the Walsh spectrum of the power permutation $f(x)=x^{d}$ over a finite field of order $2^{2m}$ and at most five distinct nonzero weights in the cyclic code of length $2^{2m}-1$ with two primitive nonzeros $\alpha $ and $\alpha ^{d}$. The method used to obtain this result proves constraints on the number of roots that certain seventh degree polynomials can have on the unit circle of a finite field. The method also works when $m$ is odd, in which case the associated crosscorrelation and Walsh spectra have at most six distinct values. [ABSTRACT FROM AUTHOR]

The binomial $B(x) = x^{3} + \beta x^{36}$ (where $\beta $ is primitive in $\mathbb {F}_{2^{2}}$) over $\mathbb {F}_{2^{10}}$ is the first known example of an Almost Perfect Nonlinear (APN) function that is not CCZ-equivalent to a power function, and has remained unclassified into any infinite family of APN functions since its discovery in 2006. We generalize this binomial to an infinite family of APN quadrinomials of the form $x^{3} + a (x^{2^{i}+1})^{2^{k}} + b x^{3 \cdot 2^{m}} + c (x^{2^{i+m}+2^{m}})^{2^{k}}$ from which $B(x)$ can be obtained by setting $a = \beta $ , $b = c = 0$ , $i = 3$ , $k = 2$. We show that for any dimension $n = 2m$ with $m$ odd and $3 \nmid m$ , setting $(a,b,c) = (\beta, \beta ^{2}, 1)$ and $i = m-2$ or $i = (m-2)^{-1} \mod n$ yields an APN function, and verify that for $n = 10$ the quadrinomials obtained in this way for $i = m-2$ and $i = (m-2)^{-1} \mod n$ are CCZ-inequivalent to each other, to $B(x)$ , and to any other known APN function over $\mathbb {F}_{2^{10}}$. [ABSTRACT FROM AUTHOR]

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 [3^m-1, 3^m-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(3^m) are used to construct optimal ternary cyclic codes with the same parameters. Nine open problems on such codes are also presented. [ABSTRACT FROM AUTHOR]

Abstract: A class of permutation polynomials with given form over finite fields is investigated in this paper, which is a further study on a recent work of Zha and Hu. Based on some particular techniques over finite fields, two results obtained by Zha and Hu are improved and new permutation polynomials are also obtained. [Copyright &y& Elsevier]

In this paper, a class of functions from Niho exponents with four-valued Walsh transform is obtained for any prime by a uniform method, and the distribution of the Walsh transform values is also completely determined. In particular, this class of functions is proven to be bent for a special case. Although it is shown that the obtained bent functions are equivalent to the Leander–Kholosha's class of bent functions, a direct and much simpler proof for the bentness of this kind of Niho functions is provided. [ABSTRACT FROM AUTHOR]

Several new classes of binary and p-ary regular bent functions are obtained in this paper. The bentness of all these functions is determined by some exponential sums over finite fields, most of which have close relations with the well-known Kloosterman sums. [ABSTRACT FROM AUTHOR]

In 2006, Helleseth and Kholosha conjectured and partially proved the existence of a class of ternary weakly regular monomial bent functions and also the expression for the dual bent function up to the sign value. The bentness was finally proved later in 2009 using a complicated technique that employs Stickelberger's theorem. Extensively using the previously found results and approaches, in this paper, a surprisingly short proof for the conjectured expression of the dual is given but without resolving the sign ambiguity. Furthermore, we resolve the sign by finding the trace representation of the dual function. [ABSTRACT FROM AUTHOR]

The Walsh transform of two-level autocorrelation sequences has played an important role in the construction of the set of sequences in which any two sequences are orthogonal. For p-ary sequences, there are only two basic classes of two-level autocorrelation sequences with no subfield structures for an arbitrary odd prime p. One is the class of p-ary m-sequences and the other is the class of p-ary Helleseth–Gong sequences. In this paper, the Walsh transform of a subclass of p-ary Helleseth–Gong sequences and the Walsh transform of their particular decimations are completely determined and are shown to be three-valued. [ABSTRACT FROM AUTHOR]

Constant composition codes are codes where the frequency distribution of the elements in a codeword is the same for all codewords. In this paper, three classes of constant composition codes are constructed. These codes are subcodes of cyclic codes which have few weights occurring among the codewords. The new codes are excellent asymptotically compared to the previously best known constant composition codes. [ABSTRACT FROM AUTHOR]

A function F from Fpn to itself is planar if for any $a\, \in \,F_{{p^n}}^*$ the function F(x+a)-F(x) is a permutation. CCZ-equivalence is the most general known equivalence relation of functions preserving planar property. This paper considers two possible extensions of CCZ-equivalence for functions over fields of odd characteristics, one proposed by Coulter and Henderson and the other by Budaghyan and Carlet. We show that the second one in fact coincides with CCZ-equivalence, while using the first one we generalize one of the known families of PN functions. In particular, we prove that, for any odd prime p and any positive integers n and m, the indicators of the graphs of functions F and F' from Fpn to Fpm are CCZ-equivalent if and only if F and F′ are CCZ-equivalent. We also prove that, for any odd prime p, CCZ-equivalence of functions from Fpn to Fpm, is strictly more general than EA-equivalence when n ≥ 3 and m is greater or equal to the smallest positive divisor of n different from 1. [ABSTRACT FROM AUTHOR]

In this paper, a simple but generic method is proposed for transforming any family of quaternary sequences, with low correlation, of any odd period N to another family of quaternary sequences of period 2N with low correlation. As an application of the generic method to sequence Family \cal A, a new optimal quaternary sequence family with length 2(2^n-1), family size 2^n+1, and maximal nontrivial correlation value 2^n+1\over 2+2, where n is an odd integer, is obtained. Most notably, unlike all the known optimal quaternary sequence families, the new family has a unique property that the odd integers 1, 3 and the even integers 0, 2 are allocated alternatively in all the sequences. [ABSTRACT FROM AUTHOR]

The p-ary function f(x) mapping GF (p4k) to GF(p) and given by f(x) = Tr4k (xp3k + p2k - pk +1 + x²) is proven to be a weakly regular bent function and the exact value of its Walsh transform coefficients is found. This is the first proven infinite class of nonquadratic generalized bent functions over the fields of an arbitrary odd characteristic. The proof is based on a few new results in the area of exponential sums and polynomials over finite fields that may also be interesting as independent problems. [ABSTRACT FROM AUTHOR]

Recently, the complete five-valued cross-correlation distribution has been determined between two m-sequences {st} and {sdt} of periods 2m - 1 that differ by the decimation d = 22k + 1/2k + 1 where m is odd and k = 1, i.e., d = 5/3. In this paper, the correlation distribution is given in terms of some exponential sums for any k when gcd(k, m) = 1 and m is odd. Furthermore, two new conjectures on exponential sums are presented that are of interest in their own right. Proving these conjectures would imply that the correlation distribution is independent of k under the conditions above and thus the same as for the case k = 1. The conjectures are proven for m odd and k = 2 and the paper gives a new result that the correlation distribution for d = 17/5 is the same as for d = 5/3. [ABSTRACT FROM AUTHOR]

In this paper, we study ternary monomial functions of the form f(x) = Trn(axd), where x Ε F3n and Trn :F3n → F3 is the absolute trace function. Using a lemma of Hou, Stickelberger's theorem on Gauss sums, and certain ternary weight inequalities, we show that certain ternary monomial functions arising in the 2006 IEEE TRANSACTIONS ON INFORMATION THEORY paper (vol. 52, pp. 2018-2032, 2006) are weakly regular bent, thus settling a conjecture of Helleseth and Kholosha. We also prove that the Coulter-Matthews bent functions are weakly regular. [ABSTRACT FROM AUTHOR]

This paper follows the recent work of Helleseth, Kholosha, Johansen, and Ness to study the cross correlation between an m-sequence of period 2m - 1 and the d-decimation of an m-sequence of a shorter period 2n - 1 for an even number m = 2n. Assuming that d satisfies d(2l + 1) = 2i mod 2n - 1) for some l > 0 and i ≥ 0, it is proved that the cross correlation takes on either exactly three or four values depending on whether l and n are coprime or not. The distribution of the cross-correlation values is also completely determined. Our results theoretically confirm the numerical data by Ness and Helleseth. It is conjectured that there are no other decimations that give at most four-valued cross correlation apart from the ones proved here. [ABSTRACT FROM AUTHOR]

Abstract: Let be the so-called classical Kloosterman sum over . In this paper, we compute modulo 24 for even , completing our previous results for odd . We extensively study the links between and other exponential sums, especially the cubic sums. We point out (as we did for odd ) that the values are involved in the computation of the weight distributions of cosets of primitive narrow sense extended BCH codes of length and minimum distance 8. We also complete some recent results on modulo 3. [Copyright &y& Elsevier]

In this paper, we present two optimal binary families of sequences of length 2n - 1 and 2(2n - 1) for odd integer n. They are obtained as the images of recently proposed optimal quaternary sequences under the most significant bit and the Gray maps. The first family has 2n + 1 sequences of length 2n - 1 and the identical correlation distribution to that of Gold sequences and Gold-like sequences, and the second family of sequences of length 2(2n -1) has 2n sequences and the same correlation values as those of Kerdock sequences. [ABSTRACT FROM AUTHOR]

Finding the cross correlation between two m-sequences {st} and {sdt} of the same period 2m - 1, that differ by a decimation d, has been a popular research problem since the 1960s. Many cases with three- and four-valued correlation have been determined. Several values of d are known to lead to five-valued cross correlation but their exact correlation distribution has been open. The correlation distribution is completely determined for one of these families with five-valued cross correlation by using evaluations of certain exponential sums including. Kloosterman sums. The decimation considered is the special decimation d = "Multiple line equation(s) cannot be represented in ASCII text" where m is odd and k = 1, i.e., d = 5/3. The paper introduces some techniques that may be useful to obtain further results on related decimations. [ABSTRACT FROM AUTHOR]

We give exact expressions for the number of coset leaders in the cosets of weight 4 of binary primitive narrow sense Bose-Chaudury-Hocquenghem (BCH) codes of length n = 2m (m even) with minimum distance 8 in terms of several exponential sums, including cubic sums and Kloosterman sums. This allows us to bound the number of coset leaders in these cosets. [ABSTRACT FROM AUTHOR]

Family A is a family of sequences of period 2n - 1 over Z4, the ring of integers modulo 4. This family has optimal correlation properties and its correlation distribution is well known. Two related families of quaternary sequences are the families B and C. These are families of sequences over Z4 of period 2 (2n - 1). In recent years, new families of quaternary sequences of period 2(2n - 1) have been constructed by modifying the sequence families B and C in a nonlinear way. This has resulted in a new family D of sequences of period 2(2n - 1) which has optimal correlation properties, but until now the correlation distribution of this family has not been known. In this paper, we completely determine the correlation distribution of family D by making use of properties of exponential sums. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, the polynomials with are studied. Some new criteria for the number of zeros of in are proved. In particular, a criterion for to have exactly one zero in when is formulated in terms of the values of polynomials introduced by Dobbertin. In the case when there is a unique zero, this root is calculated explicitly. [Copyright &y& Elsevier]

The cross-correlation function between m-sequences of period 2m -1, where m = 6k, and m-sequences of shorter period 2m/2 - 1 is investigated. The first infinite family of pairs of m-sequences with four-valued cross correlation is constructed and the complete correlation distribution of this family is determined. [ABSTRACT FROM AUTHOR]

In this correspondence, we present a connection between designing low-correlation zone (LCZ) sequences and the results of correlation of sequences with subfield decompositions presented in a recent book by the first two authors. This results in LCZ signal sets with huge sizes over three different alphabetic sets: finite field of size q, integer residue ring modulo q, and the subset in the complex field which consists of powers of a primitive qth root of unity. We show a connection between these sequence designs and ‘completely noncyclic’ Hadamard matrices and a construction for those sequences. We also provide some open problems along this direction. [ABSTRACT FROM AUTHOR]

In this paper, a coding-theory construction of Cartesian authentication codes is presented. The construction is a generalization of some known constructions. Within the framework of this generic construction, several classes of authentication codes using certain classes of error-correcting codes are described. The authentication codes presented in this paper are better than known ones with comparable parameters. It is demonstrated that the construction is related to certain combinatorial designs, such as difference matrices and generalized Hadamard matrices. [ABSTRACT FROM AUTHOR]

Considered is the distribution of the cross correlation between m-sequences of length 2m - 1, where m = 2k, and m-sequences of shorter length 2k - 1. New pairs of m-sequences with three-valued cross correlation are found and the complete correlation distribution is determined. Finally, we conjecture that there are no more cases with a three-valued cross correlation apart from the ones proven here. [ABSTRACT FROM AUTHOR]

The filter generator is an important building block in many stream ciphers. The generator consists of a linear feedback shift register of length n that generates an m-sequence of period 2′ - 1 filtered through a Boolean function of degree d that combines bits from the shift register and creates an output bit zt at any time t. The previous best attacks aimed at reconstructing the initial state from an observed keystream, have essentially reduced the problem to solving a nonlinear system of D = (Multiple line equation(s) cannot be represented in ASCII text) (i) equations in n unknowns using techniques based on linear algebra. This attack needs about D bits of keystream and the system can be solved in complexity O (Dω), where ω can be taken to be Strassen's reduction exponent ω = log2 (7) ≈ 2.807. This paper describes a new algorithm that recovers the initial state of most filter generators after observing O(D) keystream bits with complexity O((D - n)/2) ≈ O(D), after a pre-computation with complexity O(D(log2 D)³). [ABSTRACT FROM AUTHOR]

Abstract: We study the inverse permutation on the field of order by means of their component functions . We prove that the weights of derivatives of can be expressed in terms of Kloosterman sums. We are then able to compute some indicators of the propagation characteristics of σ. We can claim that σ, which is considered as a good cryptographic mapping regarding several criteria, is moreover such that the functions have good propagation properties with respect to these indicators. We further deduce several new formulas on Kloosterman sums, by using classical formulas which link any Boolean function with its derivatives. [Copyright &y& Elsevier]

Abstract: Assume that . We study the number of solutions to in . In addition, we give a simple proof of the fact that the cross-correlation function between two m-sequences, which differ by a decimation of this type, is at least four-valued. [Copyright &y& Elsevier]

Abstract: In a previous paper, we studied the cosets of weight 4 of binary extended 3-error-correcting BCH codes of length (where m is odd). We expressed the number of codewords of weight 4 in such cosets in terms of exponential sums of three types, including the Kloosterman sums , . In this paper, we derive some congruences which link Kloosterman sums and cubic sums. This allows us to study the divisibility of Kloosterman sums modulo 24. More precisely, if we know the traces of a and of , we are able to evaluate modulo 24 and to compute the number of those a giving the same value of modulo 24. [Copyright &y& Elsevier]

Linear equations have always been powerful tools in crypt-analysis. In this correspondence, we present a general linear equation of minimum weight 3 in F2 that holds for all state lengths n and all shifts i of sequences generated by the T-function xi = xi-1² V C + xi-1 rood 2n proposed by Klimov and Shamir. It is surprising that these linear properties exist, and they indicate that the sequences generated by the T-functions have more structures than claimed by Klimov and Shamir. [ABSTRACT FROM AUTHOR]

The distribution of the cross correlation between m-sequences of period 2m - 1, where m = 2k, and m-sequences of shorter period 2k - 1 is considered. A new pair of m-sequences with three-valued cross correlation is found and the complete correlation distribution is determined. [ABSTRACT FROM AUTHOR]

The distribution of the cross correlation between the ternary m-sequence {st} of period n - 3m - 1 and the decimated sequences {sdt} and {sdt + 1} of period (3m - 1)/2, where d = 3k + 1/2 with k odd and gcd(k, m) = 1 is determined. The method to find this distribution is related to the result by Coulter and Matthews that f(x) = xd is a planar function over GF(3m). [ABSTRACT FROM AUTHOR]

Considered are p-ary bent functions having the form (multiple line equation(s) cannot be represented in ASCII text). A new class of ternary monomial regular bent function with the Dillon exponent is discovered. The existence of Dillon bent functions in the general case is an open problem of deciding whether a certain Kloosterman sum can take on the value -1. Also described is the general Gold-like form of a bent function that covers all the previously known monomial quadratic cases. The (weak) regularity of the new as well as of known monomial bent functions is discussed and the first example of a not weakly regular bent function is given. Finally, some criteria for an arbitrary quadratic function to be bent are proven. [ABSTRACT FROM AUTHOR]

Binary primitive triple-error-correcting Bose--Chaudhuri-Hocquenghem (BCH) codes of length n = 2m - 1 have been the object of intensive studies for several decades. In the 1970s, their covering radius was determined in a series of papers to be ρ = 5. However, one problem for these codes that has been open up to now is to find their coset distribution. In this paper this problem is solved and the number of cosets of each weight in any binary primitive triple-error-correcting BCH code is determined. As a consequence this also gives the coset distribution of the extended codes of length N = 2m with minimum distance 8. [ABSTRACT FROM AUTHOR]

Abstract: Let be a primitive element of , where . Let , where , and s is such that divides k. We prove that the binary m-sequences and have a four-level cross-correlation function and give the distribution of the values. [Copyright &y& Elsevier]

The monotone structure of correctable and uncorrectable errors given by the complete decoding for a binary linear code is investigated. New bounds on the error-correction capability of linear codes beyond half the minimum distance are presented, both for the best codes and for arbitrary codes under some restrictions on their parameters. It is proved that some known codes of low rate are as good as the best codes in an asymptotic sense. [ABSTRACT FROM AUTHOR]

Separating codes find applications in many fields including automata theory and digital fingerprinting. It is known that the Kerdock code of sufficient order is (2, 1)- and (2, 2)-separating, but the separating weight is only known by a lower bound due to Sagalovich. In this correspondence, we prove that the lower bound on the (2, 1) -separating weight is met with equality. [ABSTRACT FROM AUTHOR]

The probability of correct decoding on the binary-symmetric channel is studied in particular, a class of codes with the same lengths and dimensions as the linear simplex codes, but with larger probability of correct decoding for all parameters p, 0 < p < 1/2, is given. [ABSTRACT FROM AUTHOR]

Recently, Helleseth, Kim, and No described the linear complexity over Fp, of Sidel'nikov sequences of length pm - 1 for p = 3,5, and 7. In this correspondence, the result is generalized to all odd primes. [ABSTRACT FROM AUTHOR]

The number of solutions of a nonlinear system of equations over the Galois field F=GF(2m) is expressed in terms of Kloosterman sums over the Galois field F. Using some natural symmetry properties of the solutions to this system, we obtain necessary and sufficient conditions for a new identity for Kloosterman sums over F. [Copyright &y& Elsevier]

Describes the linear complexity over F [sub p] and trace representation of Lempel-Cohn-Eastman (LCE) sequences. Derivation of the exact closed form expressions for the linear complexity of F [sub p] of LCE sequences; Determination of the trace representations for LCE sequences of period p; Computation of the values of all Fourier coefficients.

We find new families of nonbinary sequences of period p[SUPn] -- 1 with symbols from a finite field F[SUBp] for any prime p ≥ 3. The sequences have two-level ideal autocorrelation and are generalizations of recently found ternary sequences with ideal autocorrelation. Difference sets with parameters ??? can also be derived from these sequences in a natural way. [ABSTRACT FROM AUTHOR]

Examines the minimum distance and certain weights of the duals of binary irreducible cyclic codes. Weight distribution of the codes; Use of the cyclotomic numbers of certain order; Description of a class of double-error correcting codes.

It has been shown by Gulliver (Discrete Math. 149 (1996) 83) that there exists a ternary [98,6,63] code. But it is unknown whether or not there exists a ternary [97,6,63] code. The purpose of this paper is to prove that there is no ternary [97,6,63] code using the structure of a {267,87;5,3}-minihyper and a generator matrix of a ternary [97,6,63] code. Since n3(6,63)=97 or 98 and d3(97,6)=62 or 63, this implies that n3(6,63)=98 and d3(97,6)=62, where n3(k,d) and d3(n,k) denote the smallest value of n and the largest value of d, respectively, for which there exists an [n,k,d] code over the Galois field GF(3). [Copyright &y& Elsevier]

Presents information on a study which examined the coset weight distributions of three-error-correcting binary Z...-linear Goethals codes. Preliminary results on uniformly packed codes; Coset weight distributions of the codes.

Focuses on a study which examined the coset weight distributions of the family of Z4-linear Goethals-like codes. Types of Goethals codes; Discussion on coset weight distributions; Theorems and proofs.

Presents decimations of Welch and Niho type on ternary m-sequences with three-valued cross-correlation function. Derivation of equations; Known decimations with three-valued cross correlation; Fourth-power sum and divisibility suffice.

Gives several families of binary sequences of period N with optimal three-level autocorrelation, where N is equivalent to 2(mod 4). Problem in sequence design; Construction of sequence based on cyclotomy; Balanced sequences with optimal three-level autocorrelation.