Search

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]

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]

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]

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]

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]

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]

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]

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]

Let m and k be positive integers with m/gcd(m,k) being odd, for a\in \BBR and b\in \BBL, the exponential sum \sumx\in \BBLi^{Tr(ax+2bx^{2^{k}+1})} is studied systematically in this paper, where i=\sqrt -1, \BBR =\BBG \BBR (4,m) is a Galois ring, \BBL is the Teichmüller set of \BBR and Tr(\cdot) is the trace function from the Galois ring \BBR to \BBZ4. Through the discussions on the solutions of certain equations and the newly developed theory of \BBZ4-valued quadratic forms, the distribution of the exponential sum is completely determined. As its applications, we can determine the Lee weight and Hamming weight distributions of a class of codes \cal C^k over \BBZ4 and the correlation distribution of a quaternary sequence family \cal U^k, respectively. Furthermore, the Hamming weight distributions of the binary codes obtained from \cal C^k under the most significant bit (MSB) and Gray maps are also determined. For the MSB map sequences of \cal U^k, the nontrivial maximal correlation value is given and the correlation distribution is determined for the Gray map sequences of \cal U^k. It should be noted that the distribution of the exponential sum for the case \gcd (m,k)\ne 1 is obtained for the first time, and then the corresponding codes and sequences are novel. [ABSTRACT FROM AUTHOR]

Suppose that n = 2k is even. We study the cross-correlation function between two m-sequences for Niho type decimations d = (2k - 1)s + 1. We develop a new technique to study the value distribution of these cross-correlation functions, which makes use of Dickson polynomials. As a first application, we derive here the distribution of the six-valued cross-correlation function for s = 3 and odd k, up to a term which depends on Kloosterman sums. In addition, applying simpler methods, we prove a theorem providing Niho type decimations with four-valued cross-correlation functions and their distribution. We conjecture that the latter result actually covers all such decimations. [ABSTRACT FROM AUTHOR]

Let p be a prime, n=2m and d=3p^m-2 with m\geq 2 , and \gcd (d,p^n-1)=1 . In this paper, the correlation distribution between a p -ary m -sequence of period p^n-1 and its d -decimation sequence is investigated in a unified approach. Some results for the binary case are extended to the general case. It is shown that the problem of determining the correlation distribution for d can be reduced to that of solving two combinatorial problems related to the unit circle of the finite field \mathbb {F}_{p^{n}} . For an arbitrary odd prime p , it seems difficult to solve these two problems. However, for p=3 , by studying the weight distribution of the ternary Zetterberg code and counting the numbers of solutions of some equations over \mathbb {F}_{3^{n}} , the two problems are solved, and thus, the corresponding correlation distribution for d$ is completely determined. It is noteworthy that this is the first time that the correlation distribution for a non-binary Niho decimation has been determined since 1976. [ABSTRACT FROM PUBLISHER]

For an odd prime \(p\) and two positive integers \(m\) , \(k\) such that \(m/\gcd (k,m)\geq 3\) is odd, let \(d\) be a positive integer satisfying \(d(p^{k}+1)\equiv 2\,({\rm mod}\,p^{m}-1)\) . In this paper, the cross-correlation between a \(p\) -ary \(m\) -sequence and its \(d\) -decimated sequences is investigated, and the cross correlation distribution is completely determined. The relationship between the decimations \(d\) considered in this paper and some known ones is also studied. This paper generalizes some previous results, and also gives new decimations, which lead to low cross correlation. [ABSTRACT FROM PUBLISHER]

For a prime power q, we show that a cyclic relative difference set with parameters (qn-1/q-1, q-1, qn-2) can be constructed from a d-homogeneous function from Fqn / {0} onto Fq with difference-balanced property, where Fqn is the finite field with qn elements. This construction method enables us to construct several new cyclic relative difference sets with parameters (pn-1/pl-1, pl - 1, pn-1, pn-2t) from p-ary sequences of period pn- 1 with ideal autocorrelation property introduced by Helleseth and Gong. Using a lifting idea, other new cyclic relative difference sets can be constructed from the Helleseth-Gong (HG) sequences. Also, the 3-ranks and the trace representation of the characteristic sequences of cyclic relative difference sets from a specific class of ternary HG sequences and ternary Lin sequences are derived. [ABSTRACT FROM AUTHOR]

For an odd prime p and integers n, m, and k such that n = (2m + 1)k, a new family of p-ary sequences of period pn - 1 with optimal correlation property is constructed using the p-ary Helleseth-Gong sequences with ideal autocorrelation, where the size of the sequence family is pn. That is, the maximum nontrivial correlation value Rmax of all pairs of distinct sequences in the family does not exceed pn/2 + 1, which means the family has optimal correlation in terms of Welch's lower bound. The symbol distribution of the sequences in the family is enumerated. It Is also shown that the linear span of the sequences in the family is (m + 2)n except for the m-sequence in the family. [ABSTRACT FROM AUTHOR]