4-phase sequences with near-optimum correlation properties

Two families of four-phase sequences are constructed using irreducible polynomials over Z₄. Family A has period L =2^r-1. size L+2. and maximum nontrivial correlation magnitude C_max⩽1+√(L+1), where r is a positive integer. Family B has period L=2(2^r-1). size (L+2)/4. and C_max for complex-valued sequences. Of particular interest, family A has the same size and period as the family of binary Gold sequences. but its maximum nontrivial correlation is smaller by a factor of √2. Since the Gold family for r odd is optimal with respect to the Welch bound restricted to binary sequences, family A is thus superior to the best possible binary design of the same family size. Unlike the Gold design, families A and B are asymptotically optimal whether r is odd or even. Both families are suitable for achieving code-division multiple-access and are easily, implemented using shift registers. The exact distribution of correlation values is given for both families     

The Correlation Distribution of Quaternary Sequences of Period $2(2^n-1)$

Family A is a family of sequences of period 2ⁿ - 1 over Zi, 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.     

Generalized Kasami Sequences: The Large Set

In this correspondence, new binary sequence families F^k of period 2ⁿ-1 are constructed for even n and any k with gcd(k,n)=2 if n/2 is odd or gcd(k,n)=1 if n/2 is even. The distribution of their correlation values is completely determined. These families have maximum correlation 2^n/2+1 and family size 2^3n/2 + 2^n/2 for odd n/2 or 2^3n/2+2^n/2-1 for even n/2. The proposed families include the large set of Kasami sequences, where the k is taken as k=n/2+1.     

Optimal biphase sequences with large linear complexity derived fromsequences over Z4

New families of biphase sequences of size 2^r-1+1, r being a positive integer, are derived from families of interleaved maximal-length sequences over Z₄ of period 2(2^r-1). These sequences have applications in code-division spread-spectrum multiuser communication systems. The families satisfy the Sidelnikov bound with equality on &thetas;_max, which denotes the maximum magnitude of the periodic cross-correlation and out-of-phase autocorrelation values. One of the families satisfies the Welch bound on &thetas;_max with equality. The linear complexity and the period of all sequences are equal to r(r+3)/2 and 2(2 ^r-1), respectively, with an exception of the single m-sequence which has linear complexity r and period 2^r-1. Sequence imbalance and correlation distributions are also computed     

On coset weight distributions of the Z4-linear Goethalscodes

We study the coset weight distributions of two well-known families of codes: the three-error-correcting binary Z₄-linear Goethals codes of length N=2^m+1, m⩾3 odd, and the Z₄ -linear Goethals codes over Z₄ of length n=N/2=2^m . The hard case is the weight distributions of cosets of weight 4. To know the weight distribution of the coset of weight 4 we have to know the number of codewords of weight 4 in such a coset. Altogether, there are nine different types of cosets of weight 4. For six cases, we give the exact expressions for the number of codewords of weight 4, and for three other cases, we give such expressions in terms of Kloosterman sums     

一类由交织方式构造的二元ZCZ序列簇

2000年, Tang, Fan和Matsufuji给出(L,M,Zcz)-ZCZ序列簇的理论界为ZczL/M-1 。给定正整数n和L,本文给出一个交织ZCZ序列簇的构造算法,该算法由L条周期为L的正交序列簇生成一类(2n+1L,2L,2n-1)-ZCZ序列簇。若n2且4 |, 该类ZCZ序列簇中编号为奇数的序列与编号为偶数的序列在移位为时相关值为零。此外,选择不同的正交序列簇或不同的移位序列, 经构造算法可以生成不同的ZCZ序列簇。     

Z4上周期为2p2的四元广义分圆序列的线性复杂度

该文根据特征为4的Galois环理论,在Z₄上利用广义分圆构造出一类新的周期为2p2(p为奇素数)的四元序列,并且给出了它的线性复杂度。结果表明,该序列具有良好的线性复杂度性质,能够抗击Berlekamp-Massey (B-M)算法的攻击,是密码学意义上性质良好的伪随机序列。     

Cyclic codes and self-dual codes over F2+uF2

We introduce linear cyclic codes over the ring F₂+uF ₂={0,1,u,u¯=u+1}, where u²=0 and study them by analogy with the Z₄ case. We give the structure of these codes on this new alphabet. Self-dual codes of odd length exist as in the case of Z₄-codes. Unlike the Z₄ case, here free codes are not interesting. Some nonfree codes give rise to optimal binary linear codes and extremal self-dual codes through a linear Gray map     

A Framework for the Construction ofGolay Sequences

In 1999, Davis and Jedwab gave an explicit algebraic normal form for m!/2 - 2^h(m+1) ordered Golay pairs of length 2^mmiddot over Z₂h, involving m!/2 - 2^h(m+1) Golay sequences. In 2005, Li and Chu unexpectedly found an additional 1024 length 16 quaternary Golay sequences. Fiedler and Jedwab showed in 2006 that these new Golay sequences exist because of a "crossover" of the aperiodic autocorrelation function of certain quaternary length eight sequences belonging to Golay pairs, and that they spawn further new quaternary Golay sequences and pairs of length 2^m for m > 4 under Budisin's 1990 iterative construction. The total number of Golay sequences and pairs spawned in this way is counted, and their algebraic normal form is given explicitly. A framework of constructions is derived in which Turyn's 1974 product construction, together with several variations, plays a key role. All previously known Golay sequences and pairs of length 2^m over Z₂h can be obtained directly in explicit algebraic normal form from this framework. Furthermore, additional quaternary Golay sequences and pairs of length 2^m are produced that cannot be obtained from any other known construction. The framework generalizes readily to lengths that are not a power of 2, and to alphabets other than Z_2h .     

New designs for signal sets with low cross correlation, balanceproperty, and large linear span: GF(p) case

New designs for families of sequences over GF(p) with low cross correlation, balance property, and large linear span are presented. The key idea of the new designs is to use short p-ary sequences of period υ with the two-level autocorrelation function together with the interleaved structure to construct a set of long sequences with the desired properties. The resulting sequences are interleaved sequences of period υ². There are υ cyclically shift distinct sequences in each family. The maximal correlation value is 2υ + 3 which is optimal with respect to the Welch bound. Each sequence in the family is balanced and has large linear span. In particular, for binary case, cross/out-of-phase autocorrelation values belong to the set {1, -υ, υ + 2, 2υ + 3, -2υ - 1}, any sequence where the short sequences are quadratic residue sequences achieves the maximal linear span. It is shown that some families of these sequences can be implemented efficiently in both hardware and software     

Cyclic codes over Z4, locator polynomials, and Newton'sidentities

Certain nonlinear binary codes contain more codewords than any comparable linear code presently known. These include the Kerdock (1972) and Preparata (1968) codes that can be very simply constructed as binary images, under the Gray map, of linear codes over Z₄ that are defined by means of parity checks involving Galois rings. This paper describes how Fourier transforms on Galois rings and elementary symmetric functions can be used to derive lower bounds on the minimum distance of such codes. These methods and techniques from algebraic geometry are applied to find the exact minimum distance of a family of Z ₄. Linear codes with length 2^m (m, odd) and size 2(2^m+1-5m-2). The Gray image of the code of length 32 is the best (64, 2³⁷) code that is presently known. This paper also determines the exact minimum Lee distance of the linear codes over Z₄ that are obtained from the extended binary two- and three-error-correcting BCH codes by Hensel lifting. The Gray image of the Hensel lift of the three-error-correcting BCH code of length 32 is the best (64, 2³²) code that is presently known. This code also determines an extremal 32-dimensional even unimodular lattice     

A new family of binary pseudorandom sequences having optimalperiodic correlation properties and larger linear span

A collection of families of binary {0,1} pseudorandom sequences is introduced. Each sequence within a family has period N=2"-1, where n=2m is an even integer. There are 2^m sequences within a family, and the maximum overall (nontrivial) auto- and cross-correlation values equals 2^m+1. Thus, these sequences are optimal with respect to the Welch bound on the maximum correlation value. Each family contains a Gordon-Mills-Welch (GMW) sequence, and the collection of families includes as a special case the small set of Kasami sequences. The linear span of these sequences varies within a family but is always greater than or equal to the linear span of the GMW sequence contained within the family. Exact closed-form expressions for the linear span of each sequence are given. The balance properties of such families are evaluated, and a count of the number of distinct families of given period N that can be constructed is provided     

On algebraic decoding of the Z4-linear Goethals-likecodes

The Z₄-linear Goethals-like code of length 2^m has 2^2m+1-3m-2 codewords and minimum Lee distance 8 for any odd integer m⩾3. We present an algebraic decoding algorithm for all Z₄-linear Goethals-like codes C_k introduced by Helleseth et al.(1995, 1996). We use Dickson polynomials and their properties to solve the syndrome equations     

一类大线性复杂度二元QF序列的构造

本文构造了一类二元QF序列,计算了周期、个数、相关函数和线性复杂度。结果表明,当n为偶数时,Cmax=qn/2+1+1,Lmax=m(3n/2)m-1;当n为奇数时,Cmax=qn/2+3/2+1,Lmax=2m-1mnm-1。     

基于两族函数的低相关二元序列集构造

低相关序列集在码分多址(CDMA)扩频通信系统和密码系统中具有极其重要的作用,运用有限域上的函数族能有效地构造相关性较好的序列集.针对n≡2(mod 4)时,本文首次运用阶数为2ⁿ的有限域上的两个二次布尔函数族,构造了2^2n-2个低相关序列集;这里每个序列集包含2ⁿ+1条周期为2ⁿ-1的二元序列,其最大相关值为2^n/2+1+1.这将为通信系统和密码系统提供更多可供选择的序列集.     

Binary sequence sets with favorable correlations from differencesets and MDS codes

We propose new families of pseudorandom binary sequences based on Hadamard difference sets and MDS codes. We obtain, for p=4k-1 prime and t an integer with 1⩽t⩽(p-1)/2, a set of p^t binary sequences of period p² whose peak correlation is bounded by 1+2t(p+1). The sequences are balanced, have high linear complexity, and are easily generated     

基于二元二值序列构造四元低相关区序列集

该文基于长度为N=ML+r的具有理想二值自相关特性的二元序列,构造了一类四元低相关区序列集。得到的序列集参数如相关区长度和序列数目等在一定范围内可以灵活设定。当r0可以同时构造出多个具有相同参数的四元低相关区序列集。当r=0时得到的低相关区序列集中序列数目接近甚至达到理论界限。该文方法可以为准同步CDMA扩频通信系统提供更多的低相关区序列。     

A large class of binary ZCZ sequence families constructed by period doubling

In an Approximately Synchronized Code Division Multiple Access （AS-CDMA） communication system, a family with large number of Zero Correlation Zone （ZCZ） sequences is desired, which can satisfy the rapid increase of users. This paper presents a method to generate a （2L,2M, ZCZ＇）-ZCZ sequence family from an original （L,M,ZCZ）-ZCZ sequence family, where ZCZ＇ = ZCZ if ZCZ is even and ZCZ＇ =ZCZ - 1 if ZCZ is odd. This method can also recursively act on a ZCZ sequence family to construct a series of ZCZ sequence families with large sequence number and zero correlation zone length identical to or one less than that of original ZCZ sequences.     

A new binary sequence family with low correlation and large size

For odd n=2l+1 and an integer /spl rho/ with 1/spl les//spl rho//spl les/l, a new family S/sub o/(/spl rho/) of binary sequences of period 2/sup n/-1 is constructed. For a given /spl rho/, S/sub o/(/spl rho/) has maximum correlation 1+2/sup n+2/spl rho/-1/2/, family size 2/sup n/spl rho//, and maximum linear span n(n+1)/2. Similarly, a new family of S/sub e/(/spl rho/) of binary sequences of period 2/sup n/-1 is also presented for even n=2l and an integer /spl rho/ with 1/spl les//spl rho/相似文献   

新的具有大线性复杂度的4值低相关序列集

设正整数n、m和r满足n=4m,r=2m-1 -1,基于Niho序列集和d型函数构造了一类4值低相关序列集S(r).该序列集中序列的数目为2n,相关函数的最大边峰值2(n+2)/2+1,序列的周期为2n-1.通过Key的方法,证明了该序列集中序列线性复杂度的F界为n(2n/2-3+2).该序列集与江文峰等人构造的序列集具有相同的相关函数值和序列数目,但拥有更大的线性复杂度.