Ideal 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.

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.

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.

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.

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.

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.

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 .

New nonbinary sequences with ideal two-level autocorrelation

We find new families of nonbinary sequences of period p/sup n/-1 with symbols from a finite field F/sub p/ for any prime p/spl ges/3. The sequences have two-level ideal autocorrelation and are generalizations of previously found ternary sequences with ideal autocorrelation. Difference sets with parameters ((p/sup n/-1)/(p-1), (p/sup n-1/-1)/(p-1), (p/sup n-2/-1)/(p-1)) can also be derived from these sequences in a natural way.

New Families of Ideal 2-Level Autocorrelation Ternary Sequences From Second Order DHT

Abstract Following the work of Gong and Golomb on binary sequences, we used the technique of applying Second Order Decimation-Hadamard Transform (DHT) operater to obtain previously unknown ternary (ideal) two-level autocorrelation sequences. This process is referred to as realization. We obtained new multi-term ternary two-level autocorrelation sequences as realization of single 2-term ternary two-level autocorrelation sequence. We conjecture that for n = 2m + 1 odd, there exist m or m − 1 infinite inequivalent families of ternary two-level autocorrelation (AC) sequence which are given by four constructions. We have verified this for n = 5, 7, 9 and 11. Experimental results are provided.

A New Family of Ternary Sequences with IdealTwo-level Autocorrelation Function

Let α be a primitive element of F_{3^n}. Let d=3^{2k}-3^k+1 where n=3k. We show that the ternary sequence \{s(t)\} given by s(t)=Tr_n(\alpha^t+\alpha^{dt}) has a two-level ideal autocorrelation function.