Two-level Autocorrelation Function Research Articles

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.

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.

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.

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 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.

Cyclic inequivalence of cascaded GMW-sequences

Cascaded GMW sequences have two-level autocorrelation functions, which have important applications in communications and cryptology. In this paper, we consider the cascaded GMW sequences corresponding to a fixed finite chain of finite fields, and determine whether the different cascaded GMW sequences are cyclically inequivalent. By introducing the so-called restricted integer systems (RISs), it is proved that all the cascaded GMW sequences can be determined by means of the RISs, and the sequences determined by different RISs are different. Moreover, different cascaded GMW sequences are cyclically inequivalent.

Perfect and almost perfect sequences

The autocorrelation function of a sequence is a measure for how much the given sequence differs from its translates. Periodic binary sequences with good correlation properties have important applications in various areas of engineering. In particular, one needs sequences with a two-level autocorrelation function, that is, all nontrivial autocorrelation coefficients equal some constant γ. The case where γ is as small as theoretically possible in absolute value has turned out to be especially useful; such sequences are called perfect. Unfortunately, in many cases perfect sequences cannot exist, and so one has also considered “almost perfect” sequences, where one allows one nontrivial autocorrelation coefficient to be different from γ. In this paper, we concentrate on the existence problem for perfect and almost perfect binary periodic sequences; such sequences are actually equivalent to certain cyclic difference sets and cyclic divisible difference sets, respectively, structures which have been studied in Design Theory for a long time. This connection allows one to obtain strong results on the existence of (almost) perfect sequences. We also discuss some related questions, namely the aperiodic case, ternary perfect sequences, binary perfect arrays, and the merit factor of a sequence. Finally, we conclude the paper with a simplified description of an interesting real-world application, namely “coded aperture imaging”. Our paper serves mainly as a survey of the area just outlined, but it also contains some new results.

On balanced binary sequences with two-level autocorrelation functions

Recently, Maschietti (Des., Codes Cryptogr., vol.14, p.89, 1998) constructed several families of (2/sup m/-1,2/sup m-1/-1,2/sup m-2/-1) cyclic difference sets from monomial hyperovals. In this correspondence, we consider the binary sequences associated to the difference sets constructed by Maschietti. We give algebraic proof of the fact that these sequences have two-level autocorrelation functions, also we discuss the linear spans of these sequences.

Cross-correlation of p-ary GMW sequences

The cross-correlation function of p-ary (p prime) GMW sequences is investigated. The well-known m-sequences are incorporated in these sequences. GMW sequences also have the two-level autocorrelation function, but their linear span may be much larger than those of m sequences. The calculation of the cross-correlation function of GMW sequences is reduced to the cross-correlation of the m-sequences because many results on this kind of sequence are known. In the literature, only the cross-correlation function of an m sequence with a GMW sequence has been known up to now. >

Complex sequences over GF(p/sup M/) with a two-level autocorrelation function and a large linear span

New complex sequences with elements are proposed that have constant absolute values of 1. The periodic autocorrelation functions of these sequences are shown to be two-level. The sequences are generated by three consecutive mapping processes. The number of sequences of fixed length is determined, which is larger than the number of binary sequences. For cryptographical reasons, the large linear span of the sequences is of great interest. The linear span of the new sequences is examined and is proven to be larger than the linear span of complex m-sequences, if the parameters of the mapping processes are appropriately chosen. The formula for generating the sequences is generalized to enlarge the linear span of sequences.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>