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In this paper, for each of six families of three-valued $ m $-sequence correlation, we construct an infinite family of five-weight codes from trace codes over the ring $ R = \mathbb{F}_2+u\mathbb{F}_2 $, where $ u^2 = 0. $ The trace codes have the algebraic structure of abelian codes. Their Lee weight distribution is computed by using character sums. Their support structure is determined. An application to secret sharing schemes is given. The parameters of the binary image are $ [2^{m+1}(2^m-1),4m,2^{m}(2^m-2^r)] $ for some explicit $ r. $ [ABSTRACT FROM AUTHOR]

As numerous applications in wireless communications and radar sensing all rely on the finite and precious spectral resource, contiguous spectrum allocation schemes have become very difficult to continue nowadays. Spectrally constrained sequences (SCSs) are specially designed sequences which display low correlation sidelobes to effectively utilize the increasingly congested and fragmented spectrum. Recently, Liu et. al (IEEE Trans. Inf. Theory 64(4):2571-2582, 2018) proposed a lower bound of the maximal correlation for SCSs and constructed two classes of optimal SCSs by using Singer difference sets and perfect ternary sequences. For the application of SCSs, it is desirable that the spectrally constrained position should be as flexible as possible, and the correlation tolerance should be as small as possible. In this paper, we present some systematic constructions based on cyclotomy which generate some asymptotically optimal SCSs with respect to Liu's bound. The proposed constructions result in SCSs with new parameters and are more flexible in terms of the 'position of the null constraint'. In addition, we propose a framework which control the power of new SCSs while maintaining the correlation magnitudes by using cyclic algorithm-new (CAN) algorithm. [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]

Quasi-complementary sequence sets (QCSSs) play an important role in multi-carrier code-division multiple-access (MC-CDMA) systems. They can support more users than perfect complementary sequence sets in MC-CDMA systems. It is desirable to design QCSSs with good parameters that are a trade-off of large set size, small periodic maximum magnitude correlation and small alphabet size. The main results are to construct new infinite families of QCSSs that all have small alphabet size and asymptotically optimal periodic maximum magnitude correlation. In this paper, we propose three new constructions of QCSSs using additive characters over finite fields. Notably, these QCSSs have new parameters and small alphabet sizes. Using the properties of characters and character sums, we determine their maximum periodic correlation magnitudes and prove that these QCSSs are asymptotically optimal with respect to the lower bound. [ABSTRACT FROM AUTHOR]

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. [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]

In this paper, let n=2k and d=3\cdot 2^{k}-2 with k\geq 3 and \gcd (d,2^n-1)=1 . Based on some analysis of certain equations over finite fields and the number of codewords with Hamming weight five in Zetterberg code, the correlation distribution between a binary m -sequence of period 2^n-1 and its $d$ -decimation sequence is completely determined. This solves a ten-year-old open problem proposed by Dobbertin et al. [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]

In 1998, Lin presented a conjecture on a class of ternary sequences with ideal two-level autocorrelation. Those sequences have a very simple structure, i.e., their trace representation has two trace monomial terms. In this paper, we present a proof for the conjecture. The mathematical tools employed are the second-order multiplexing decimation-Hadamard transform, Stickelberger’s theorem, the Teichmüller character, and combinatorial techniques for enumerating the Hamming weights of ternary numbers. As a by-product, we also prove that the ternary sequences conjectured by Lin are Hadamard equivalent to ternary \(m\) -sequences. [ABSTRACT FROM PUBLISHER]

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]

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]

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]