Two-level Autocorrelation Research Articles

Constructions of Complementary Sequence Sets and Complete Complementary Codes by Ideal Two-Level Autocorrelation Sequences and Permutation Polynomials

In this paper, we further investigate the constructions of complementary sequence sets (CSSs) and complete complementary codes (CCCs) by Butson-type Hadamard matrices. By taking the algebraic structure of Butson-type Hadamard (BH) matrices into consideration, we obtain the explicit representation of the δ-linear terms and δ-quadratic terms, which are ingredients to construct CSSs and CCCs. In particular, we derive the δ-quadratic terms determined by DFT matrices and BH matrices constructed from 2-level autocorrelation sequences, which yields two type of new contructions. We show that inequivalent BH matrices produce different CSSs and CCCs, which proves that our constructed CSSs and CCCs are new. As a consequence of the first type of the constructions, not only a large number of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</i> -ary CSSs and CCCs of size <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</i> ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</i> prime) have been proposed, which were never reported in the literature, but also a theory linking these CSSs of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</i> -ary sequences and the generalized Reed-Muller codes proposed by Kasami <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">et al</i> . is shown. These codes enjoy good error-correcting capability, tightly controlled PMEPR, and significantly extend the range of coding options for applications of OFDM using <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">pⁿ</i> subcarriers. As a consequence of the second type of the constructions, we reveal an extremely fascinating hidden connection between the sequences in aperiodic CSSs and CCCs and the sequences with ideal period 2-level autocorrelation, through their trace representations and permutation polynomials over finite fields.

Algorithm for Generating Extra-Long Gordon–Mills–Welch Sequences

The modification of an algorithm for determining polynomial multipliers hci(x) of test polynomial hGMWS(x), which is the main component of the method for synthesizing Gordon–Mills–Welch sequences (GMWSs), is set as a basis for developing the software implementation of the algorithm for generating extra-long GMWSs having a two-level periodic autocorrelation function and high structural latency and generated over the finite field with double extension GF(2S) = GF[(2m)n]. In the modified algorithm, the expressions for the number of operations in computing the vector of alternatives, from which the vector of decimation indices is formed, are derived for different values of parameter n. These expressions also serve as upper estimates for the number of summed sequences. To generate extra-long GMWSs with periods from N = 212 – 1 = 4095 to N = 220 – 1 = 1 048 575, the sets of vectors of decimation indices are obtained for admissible values of parameters m and n.

A Note on Hall’s Sextic Residue Sequence: Correlation Measure of Order $k$ and Related Measures of Pseudorandomness

It is known that Hall’s sextic residue sequence has some desirable features of pseudorandomness: an ideal two-level autocorrelation and linear complexity of the order of magnitude of its period $p$ . Here we study its correlation measure of order $k$ and show that it is, up to a constant depending on $k$ and some logarithmic factor, of order of magnitude $p^{1/2}$ , which is close to the expected value for a random sequence of length $p$ . Moreover, we derive from this bound a lower bound on the $N$ th maximum order complexity of order of magnitude $\log p$ , which is the expected order of magnitude for a random sequence of length $p$ .

On the 2-Adic Complexity of the Two-Prime Generator

Hu introduced a simple method to compute the 2-adic complexity of any periodic binary sequence with ideal two-level autocorrelation. We extend this approach to some other sequences. First, we provide a substantially shorter proof of the maximality of the 2-adic complexity of the Legendre sequence of period $N\equiv 1\bmod 4$ first proved by Xiong et al . Then, we show that the 2-adic complexity of the two-prime generator of period $pq$ with two odd primes $p\ne q$ attains the maximum if $(q+1)/4\le p\le 4q-1$ . This result was only known for twin primes $q=p+2$ before. For arbitrary odd primes $p\ne q$ , we can still prove that the 2-adic complexity of the two-prime generator is close to its period.

A Proof of Turyn's Conjecture: Nonexistence of Circulant Hadamard Matrices for Order Greater than Four

Biphase periodic sequences having elements +1 or -1 with the two-level autocorrelation function are desirable in communications and radars. However, in case of the biphase orthogonal periodic sequences, Turyn has conjectured that there exist only sequences with period 4, i.e., there exist the circulant Hadamard matrices for order 4 only. In this paper, it is described that the conjecture is proved to be true by means of the isomorphic mapping, the Chinese remainder theorem, the linear algebra, etc.

Character Sum Factorizations Yield Sequences With Ideal Two-Level Autocorrelation

We give a new existence criterion for $p$ -ary sequences which have ideal two-level autocorrelation; and we use it to obtain four general families of such sequences: one for $p=2$ , one for general odd primes $p$ and two special ones for $p=3$ . The binary family turns out to be equivalent to that discovered by Dillon and Dobbertin and published in 2004. The general $p$ -ary family is equivalent to that discovered by Gong and Helleseth, by Dillon and, when $p=3$ , by Helleseth, Kumar, and Martinsen. All of these $p$ -ary results were published in 2001 and 2002. The special ternary families are new and give as special cases the sequences conjectured by Alfred Lin in his 1998 Ph.D. thesis as well as most of those conjectured in 2001 by Ludkovski and Gong. Our sequences may also be used to construct (relative) difference sets, their corresponding block designs and generalized weighing matrices.

Difference balanced functions and their generalized difference sets

Difference balanced functions from Fqn⁎ to Fq are closely related to combinatorial designs and naturally define p-ary sequences with the ideal two-level autocorrelation. In the literature, all existing such functions have the homogeneous property, and it was conjectured by Gong and Song that difference balanced functions must be d-homogeneous for some d with gcd⁡(d,q−1)=1. Here we characterize difference balanced functions by difference sets with respect to two exceptional subgroups. We then reveal an unexpected equivalence between the homogeneous property and certain multipliers of the corresponding difference sets. By determining these multipliers, we prove the Gong–Song conjecture for q prime.

The Constructions of Ternary Sequence Pairs with Two-Level Autocorrelation

In this letter, a periodic autocorrelation signal is presented, which is the ternary sequence pair with two-level autocorrelation. The methods of constructing ternary sequence pairs based on binary sequence pairs and cyclotomy are presented by us. Then we have a discussion about perfect binary sequence pairs, pseudorandom binary sequence pairs, difference set pairs and ternary sequence pairs. These signals enrich the theory of sequence pair discrete correlation signals.

Comments on “A New Method to Compute the 2-Adic Complexity of Binary Sequences”

We show that there is a very simple approach to determine the 2-adic complexity of periodic binary sequences with ideal two-level autocorrelation. This is the first main result by H. Xiong, L. Qu, and C. Li, IEEE Transactions on Information Theory, vol. 60, no. 4, pp. 2399-2406, Apr. 2014, and the main result by T. Tian and W. Qi, IEEE Transactions on Information Theory, vol. 56, no. 1, pp. 450-454, Jan. 2010.

The Proof of Lin’s Conjecture via the Decimation-Hadamard Transform

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 Teichmuller 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.

New Families of Binary Sequence Pairs With Two-Level and Three-Level Correlation

A pair of binary sequences is generalized from the concept of a two-level autocorrelation function of a single binary sequence. In this paper, new families of binary sequence pairs with period N = np, where gcd(n,p) = 1, and optimal correlation values -1 are constructed, as well as new families of binary sequence pairs with three-level correlation and period N≡0(mod 4). For the new binary sequence pairs with optimal correlation values, their corresponding new difference set pairs and almost difference set pairs are also derived.

Method of Generating Multiple Sets of Orthogonal Codes with Wide Choice of Spreading Factors

This paper presents a generic method to construct multiple sets of orthogonal codes for application in direct sequence spread spectrum systems. The plurality of orthogonal sets is generated using any single two-level autocorrelation sequence of period one less than the desired spreading factor of the orthogonal codes. The proposed method hence generates orthogonal codes of all spreading factors for which a corresponding two-level autocorrelation sequence exists. The resulting orthogonal sets with wide choice of spreading factors is desirable in synchronous direct sequence spread spectrum systems. The sets can be generated such that they do not contain the codeword with single polarity as in the case of Walsh-Hadamard codes and orthogonal sets generated from cyclic shifts of two-level autocorrelation sequences. This renders all codes in the generated sets with an effective spreading gain. The generation method also encompasses a procedure to generate N-1 sets with low inter-set correlation for application to oversaturated spread spectrum systems.

A Note on "A Novel ZCZ Code Based on m-Sequences and its Applications in CDMA Systems"

This short communication shows that zero correlation zone sequences could be generated by using any odd-length binary sequence with ideal two-level autocorrelation function in place of the m-sequence using the exact same procedure proposed in the original paper. This removes the restriction on the spreading factor of the m-ZCZ sequences which could assume only the form 2n for some integer n. The spreading factor can now be any N for which an ideal two-level autocorrelation sequence of period N - 1 exists.

A Three-Valued Walsh Transform From Decimations of Helleseth–Gong Sequences

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.

New Method to Extend the Number of Quaternary Low Correlation Zone Sequence Sets

In this correspondence, a new method to extend the number of quaternary low correlation zone (LCZ) sequence sets is presented. Based on the inverse Gray mapping and a binary sequence with ideal two-level auto-correlation function, numbers of quaternary LCZ sequence sets can be generated by choosing different parameters. There is at most one sequence cyclically equivalent in different LCZ sequence sets. The parameters of LCZ sequence sets are flexible.

On Binary Sequence Pairs with Two-Level Periodic Autocorrelation Function

Binary sequence pairs as a class of mismatched filtering of binary sequences can be applied in radar, sonar, and spread spectrum communication system. Binary sequence pairs with two-level periodic autocorrelation function (BSPT) are considered as the extension of usual binary sequences with two-level periodic autocorrelation function. Each of BSPT consists of two binary sequences of which all out-phase periodic crosscorrelation functions, also called periodic autocorrelation functions of sequence pairs, are the same constant. BSPT have an equivalent relationship with difference set pairs (DSP), a new concept of combinatorial mathematics, which means that difference set pairs can be used to research BSPT as a kind of important tool. Based on the equivalent relationship between BSPT and DSP, several families of BSPT including perfect binary sequence pairs are constructed by recursively constructing DSP on the integer ring. The discrete Fourier transform spectrum property of BSPT reveals a necessary condition of BSPT. By interleaving perfect binary sequence pairs and Hadamard matrix, a new family of binary sequence pairs with zero correlation zone used in quasi-synchronous code multiple division address is constructed, which is close to the upper theoretical bound with sequence length increasing.

WG: A family of stream ciphers with designed randomness properties

In this paper we present a family of stream ciphers which generate a keystream with ideal two-level autocorrelation. The design also guarantees other randomness properties, i.e., balance, long period, ideal tuple distribution, and high and exact linear complexity. We discuss how these properties are achieved by the proposed design and show how to select various parameters to obtain an efficient stream cipher for the desired security level. We also show that the proposed generators are secure against time/memory/data tradeoff attacks, algebraic attacks and correlation attacks. Finally we present WG-128 1 WG-128 is a slightly modified version of the WG stream cipher which is a phase 2 candidate in profile 2 of the ECRYPT stream cipher project: eSTREAM. See http://www.ecrypt.eu.org/stream/ 1 as a concrete example of a WG stream cipher with a key size of 128 bits.

Two-tuple balance of non-binary sequences with ideal two-level autocorrelation

Let p be a prime, q = p m and F q be the finite field with q elements. In this paper, we will consider q-ary sequences of period q n - 1 for q > 2 and study their various balance properties: symbol-balance, difference-balance, and two-tuple-balance properties. The array structure of the sequences is introduced, and various implications between these balance properties and the array structure are proved. Specifically, we prove that if a q-ary sequence of period q n - 1 is difference-balanced and has the “cyclic” array structure then it is two-tuple-balanced. We conjecture that a difference-balanced q-ary sequence of period q n - 1 must have the cyclic array structure. The conjecture is confirmed with respect to all of the known q-ary sequences which are difference-balanced, in particular, which have the ideal two-level autocorrelation function when q = p .

Cryptographic properties of the Welch-Gong transformation sequence generators

Welch-Gong (WG) transformation sequences are binary sequences of period 2/sup n/ - 1 with two-level autocorrelation. These sequences were discovered by Golomb, Gong, and Gaal (1998) and they verified the validity of their construction for 5 /spl les/ n /spl les/ 20. Later, No, Chung, and Yun (1998) found another way to construct the WG sequences and verified their result for 5 /spl les/ n /spl les/ 20. Dillon (1998) first proved this result for odd n, and, finally, Dobbertin and Dillon (1999) proved it for even n. In this paper, we investigate a two-faced property of the WG transformation sequences for application in stream ciphers and pseudorandom number generators. One is to present the randomness or unpredictability of the WG transformation sequences. The other is to exhibit the security properties of the WG transformations regarded as Boolean functions. In particular, we prove that the WG transformation sequences, in addition to the known two-level autocorrelation and three-level cross correlation with m-sequences, have the ideal 2-tuple distribution, and large linear span increasing exponentially with n. Moreover, it can be implemented efficiently. This is the first type of pseudorandom sequences with good correlation, statistic properties, large linear span, and efficient implementation. When WG transformations are regarded as Boolean functions, they have high nonlinearity. We derive a criterion for the Boolean representation of WG transformations to be r-resilient and show that they are at least 1-resilient under some basis of the finite field GF (2/sup n/). An algorithm to find such bases is given. The degree and linear span of WG transformations are presented as well.

New designs for signal sets with low cross correlation, balance property, 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 /spl upsi/ 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 /spl upsi//sup 2/. There are /spl upsi/ cyclically shift distinct sequences in each family. The maximal correlation value is 2/spl upsi/ + 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, -/spl upsi/, /spl upsi/ + 2, 2/spl upsi/ + 3, -2/spl upsi/ - 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.