Large Linear Span Research Articles

Linear Span of the Optimal Frequency Hopping Sequences from Irreducible Cyclic Codes

Optimal set of the frequency hopping sequences can be derived from some irreducible cyclic codes. This paper determines the linear span of the frequency hopping sequences in the optimal set. The linear span is much less than the length of the frequency hopping sequences. In order to improve the linear span, we use two types of permutation polynomials, power permutation and binomial permutation, to transform the optimal set of the frequency hopping sequences. The transformed frequency hopping sequences have optimal Hamming correlation and larger linear span than the original frequency hopping sequences. Compared with the original frequency hopping sequences, the transformed optimal frequency hopping sequences have higher security to resist the cryptanalytic method.

A Class of Large Family of Binary Sequences with Low Correlation and Large Linear Span

In this paper, a new family of binary sequences of period is proposed, where and . The presented family takes 7-valued correlation values , , , , , and . For and , it is proved that the proposed sequence family has linear spans , where l = 2, 3, 4, 5, 6, 7, and the distribution of linear span of sequences in is determined.

Linear spans of optimal sets of frequency hopping sequences

Frequency hopping sequences sets are required in frequency hopping code division multiple access systems. For the anti-jamming purpose, frequency hopping sequences are required to have a large linear span. In this paper, by using a permutation polynomial δ (x ) over a finite field, we transform several optimal sets of frequency hopping sequences with small linear span into ones with large linear span. The exact values of the linear span are presented by using the methods of counting the terms of the sequences representations. The results show that the transformed frequency hopping sequences are optimal with respect to the Peng-Fan bound, and can resist the analysis of Berlekamp-Massey algorithm.

Analysis of Cross-Correlation of Extended Non-Linear Binary Sequences

Code-Division Multiple-Access(CDMA) allows several users simultaneous access to a common channel by assigning a distinct pseudonoise sequence called spectrum code to each user. Each user in a CDMA system uses a assigned spectrum code to modulate their signal. Choosing the codes used to modulate the signal is very important in the performance of CDMA systems. The best performance will occur when there is good separation between the signal of a desired user and the signals of other users. The receiver synchronizes the code to recover the data. The use of an independent code allows multiple users to access the same frequency band at the same time. In this paper we propose a generalized model of non-linear binary sequence using trace function and analyze cross-correlation of these sequences. These sequences with low correlation, large linear span and large family size, in a direct-sequence spread spectrum communication system, help to minimize multiple access interference, increase security degree of system and enlarge user number.

A family of quadriphase sequences of period 4(2 n − 1) with low correlation and large linear span

In this paper, a new family of quadriphase sequences with period 4(2 n − 1) is proposed for n = me, where m is an odd integer. The correlation values of the family, their distribution, and the linear spans of the proposed sequences are completely determined under two situations.

New p-ary sequence family with low correlation and large linear span

In this paper, for an odd prime p and positive integers n, m, and e such that n = me, a new family $${\mathcal{S}}$$ of p-ary sequences of period p n − 1 with low correlation and large linear span is constructed. It is shown that $${\mathcal{S}}$$ has maximum correlation $${1+p^{n+2e\over 2}}$$, family size p n , and maximal linear span $${{(m+3)n\over 2}}$$. When m is even, the proposed family $${\mathcal{S}}$$ contains Tang, Udaya, and Fan’s construction as a subset. Furthermore, when n is even and $${e=1, \mathcal{S}}$$ has the same correlation and family size, but larger linear span compared with the construction by Seo, Kim, No, and Shin.

New nonbinary sequence families with low correlation, large size, and large linear span

Let p be an odd prime and n = ( 2 m + 1 ) e . Based on the theory of quadratic forms over finite fields of odd characteristic, we generalize the binary construction by Yu and Gong to p -ary case. As a result, we obtain a new family F o k ( ρ ) of p -ary sequences of period p n − 1 for arbitrary positive integers 1 ≤ ρ ≤ m and k with gcd ( n , k ) = e . It is shown that, for a given ρ , F o k ( ρ ) has family size p n ρ , maximum correlation 1 + p n + ( 2 ρ − 1 ) e 2 , and maximum linear span ( m + 1 ) n . In particular, the new family F o k ( ρ ) contains Tang, Udaya, and Fan’s construction as a subset, if an m -sequence is excluded.

The linear span of the frequency hopping sequences in optimal sets

Frequency hopping (FH) sequences are needed in FH code division multiple access (CDMA) systems. Recently some new constructions of optimal sets of FH sequences were presented. For the anti-jamming purpose, FH sequences are required to have a large linear span. The objective of this paper is to determine both the linear spans and the minimal polynomials of the FH sequences in these optimal sets. Furthermore, the linear spans of the transformed FH sequences by applying a power permutation are also investigated. If the power is chosen properly, the linear span could be very large compared to the length of the FH sequences.

Optimal Sets of Frequency Hopping Sequences With Large Linear Spans

Frequency hopping (FH) is one of the basic spread coding technologies in spread spectrum communications. FH sequences are needed in FH code-division multiple access (CDMA) systems. For the anti-jamming purpose, FH sequences are required to have a large linear span. A few optimal sets of FH sequences are available in the literature. However, their sequences have very small linear spans. It is known that an optimal set of FH sequences could be transformed to another optimal set of FH sequences with large linear spans by a power permutation, if the power is chosen properly [see C. Ding and J. Yin, IEEE Trans. Inf. Theory, vol. IT-54, pp. 3741-3745, 2008]. The objective of this paper is to investigate this idea of C. Ding and J. Yin further, and determine the linear span of the FH sequences in the optimal sets obtained by applying a power permutation to some existing optimal sets of FH sequences.

A Family of Binary Sequences with 4-Valued Low Correlation and Large Linear Span

A Family of Binary Sequences with 4-Valued Low Correlation and Large Linear Span

A Family of Binary Sequences with 4-Valued Low Correlation and Large Linear Span

For n≡0mod 4,based on d-form function,a new family of binary sequences with period 2n-1 and four-valued low correlation is proposed.The correlation distribution of the proposed family is completely determined.The linear spans of the new sequences are proved to be large and their exact values are also obtained.Compared with the known sequence families,the new family has not only low correlation,but also much larger linear spans.This family of sequences is suitable for cryptography and CDMA systems.

New Optimal Quadriphase Sequences With Larger Linear Span

In this paper, two new optimal families S and U of quadriphase sequences are presented. Compared to the family A constructed by Boztas and the family D investigated by Tang respectively, the proposed families have the same optimal correlation properties and family size, but larger linear spans.

New Construction of Generalized Hadamard Matrices

Based on trace function over finite field GF(pn), new construction of generalized Hadamard matrices with order pn is presented, where p is prime and n is even. The rows in new generalized Hadamard matrices are cyclically distinct and have large linear span, which greatly improves the security of the system employing them as spreading sequences.

New Families of Binary Sequences with Low Correlation and Large Size

In this paper, for odd n and any k with gcd(n, k) = 1, new binary sequence families Sk of period 2n-1 are constructed. These families have maximum correlation $1+2^{n+3\\over 2}$, family size 22n+2n+1 and maximum linear span $n(n+1)\\over 2$. The correlation distribution of Sk is completely determined as well. Compared with the modified Gold codes with the same family size, the proposed families have the same period and correlation properties, but larger linear span. As good candidates with low correlation and large family size, the new families contain the Gold sequences and the Gold-like sequences. Furthermore, Sk includes a subfamily $\\mathcal{S}^k_1$ which has the same period, correlation distribution, family size and linear span as the family So(2) recently constructed by Yu and Gong. In particular, when k=1, $\\mathcal{S}^k_1$ is exactly So(2).

Several Families of Sequences with Low Correlation and Large Linear Span

In DS-CDMA systems and DS-UWB radios, low correlation of spreading sequences can greatly help to minimize multiple access interference (MAI) and large linear span of spreading sequences can reduce their predictability. In this letter, new sequence sets with low correlation and large linear span are proposed. Based on the construction Tr1m[Trmn(αbt + γiαdt)]r for generating p-ary sequences of period pn-1, where n = 2m, d = upm ± v, b = u ± v, γi e GF(pn), and p is an arbitrary prime number, several methods to choose the parameter d are provided. The obtained sequences with family size pn are of four-valued, five-valued, six-valued or seven-valued correlation and the maximum nontrivial correlation value is (u + v-1)pm-1. The simulation by a computer shows that the linear span of the new sequences is larger than that of the sequences with Niho-type and Welch-type decimations, and similar to that of [10].

On the linear complexity of some new q-ary sequences

Some new q-ary sequences with period q 3 ek - 1 ( q = p m , p an odd prime, m, e, k integers) are first constructed and then, inspired by Antweiler’s method, their linear complexity is examined. The exact value of linear complexity k ( 6 e ) w is determined when r = ∑ i = 1 w p e i . Furthermore, an upper bound of the linear complexity is given for the other values of r. Our results show that this sequence has larger linear span than GMW sequence with the same parameters. Finally, the results of a Maple program are included to illustrate the validity of the results.

On the Linear Span of a Binary Sequence Family with Optimal Correlation Properties

A new subfamily of sequences with optimal correlation properties is constructed for the generalized Kasami set. A lower bound on the linear span is established. It is proved that with suitable choices of parameters, this subfamily has exponentially larger linear spans than either No sequences or TN sequences. A class of sequences with ideal autocorrelation is also proved to have large linear span.

New families of p-ary sequences with low correlation and large linear span

This article presents a new family of p-ary sequences. The proposed sequences are proved to have not only low correlation property, but also large linear span and large family size. Furthermore, it shows that the new family of sequences contains Tang's construction as a subset if m-sequences are excluded from both constructions.

New Construction for Balanced Boolean Functions with Very High Nonlinearity

In the past twenty years, there were only a few constructions for Boolean functions with nonlinearity exceeding the quadratic bound 2n-1 - 2(n-1)/2 when n is odd (we shall call them Boolean functions with very high nonlinearity). The first basic construction was by Patterson and Wiedemann in 1983, which produced unbalanced function with very high nonlinearity. But for cryptographic applications, we need balanced Boolean functions. Therefore in 1993, Seberry, Zhang and Zheng proposed a secondary construction for balanced functions with very high nonlinearity by taking the direct sum of a modified bent function with the Patterson-Wiedemann function. Later in 2000, Sarkar and Maitra constructed such functions by taking the direct sum of a bent function with a modified Patterson-Wiedemann function. In this paper, we propose a new secondary construction for balanced Boolean functions with very high nonlinearity by recursively composing balanced functions with very high nonlinearity with quadratic functions. This is the first construction for balanced function with very high nonlinearity not based on the direct sum approach. Our construction also have other desirable properties like high algebraic degree and large linear span.

Structure of frequency hopping sequences family based on block cipher

A novel family of frequency-hopping (FH) sequences based on iterated block cipher is proposed for frequencyhopping multiple-access (FHMA) communications. The design offers a class of nonlinear FH codes with high security, large linear span and a uniform spread over the entire frequency bandwidth. Moreover, FH sequences among the family are independent from each other and they perform as well as random patterns in terms of multiple access interference in anti-jamming applications. With the performance of packet error and throughput for FHMA network being derived in theory, many numerical results of the 3DES sequences are presented, comparing with those of shift register sequences and chaotic FH sequences. Efficiently implemented in field programmable gate arrays (FPGA), the generator prototype of the proposed sequence has been realized and incorporated into fast FH radio.