Search

We present a new construction of a family of perfect ternary sequences (PTSs) that is a generalization of one of the known families of PTSs. These PTSs of length N1N2 are derived from shift sequences of odd length N1 corresponding to m-sequences over GF(p) and PTSs of odd length N2. Ipatov PTSs are a special case where N2 = 1. For N2 ≥ 3, we find conditions under which the obtained PTSs are new. We also consider implementation issues of these sequences.

The paper describes the construction and properties of a new class of sequences. The construction relies on the decomposition of m-sequences into arrays where each column is a cyclic shift of a shorter m-sequence or constant. The m-sequence columns are replaced by suitable sequences of the same length. Three examples of such substitution are presented: complex Legendre sequence, modified Zech logarithm sequence, and Walsh-Hadamard sequences. Constant columns are replaced by all 0s (magnitude). The arrays are then unfolded to form long sequences. These sequences have a large zero correlation region around the autocorrelation peak. Many large sets of these sequences can be constructed, such that their cross-correlation is also zero in this region. Such sets are required for quasi-synchronous CDMA.

New binary and ternary sequences with low correlation and simple implementation are presented. The sequences are unfolded from arrays, whose columns are cyclic shifts of a short sequence or constant columns and whose shift sequence (sequence of column shifts) has the distinct difference property. It is known that a binary m-sequence/GMW sequence of length 2$^{\rm 2{\it m}}$ – 1 can be folded row-by-row into an array of 2m – 1 rows of length 2m + 1. We use this to construct new arrays which have at most one column matching for any two dimensional cyclic shift and therefore have low off-peak autocorrelation. The columns of the array can be multiplied by binary orthogonal sequences of commensurate length to produce a set of arrays with low cross-correlation. These arrays are unfolded to produce sequence sets with identical low correlation.

Ternary (–1, 0, +1) almost-perfect and odd-perfect autocorrelation sequences are applied in many communication, radar and sonar systems, where signals with good periodic autocorrelation are required. New families of almost-perfect ternary (APT) sequences of length N = (pn – 1)/r, where r is an integer, and odd-perfect ternary (OPT) sequences of length N/2, derived from the decomposition of m-sequences of length pn – 1 over GF(p), n = km, with p being an odd prime, into an array with T = (pn – 1)/( pm – 1) rows and pm – 1 columns, are presented. In particular, new APT sequences of length 4(pn – 1)/(pm – 1), (pm + 1) = 2 mod 4, and OPT sequences of length 2(pn – 1)/(pm – 1) with peak factor close to 1 when p becomes large, are constructed. New perfect 4-phase and 8-phase sequences with some zeroes can be derived from these OPT sequences. The obtained APT sequences of length 4(pm + 1), p > 3 and m – any even positive integer, possess length uniqueness in comparison with known almost-perfect binary sequences.

Sequences and arrays with good auto and cross-correlation are useful in communications and digital watermarking respectively. Two-valued autocorrelation is relatively easy to achieve. However, sets of pseudonoise sequences with good cross-correlation are small and difficult to find. Here, the cross-correlation for particular shifts of m-sequences and GMW sequences is analysed by looking at these sequences as two-dimensional arrays. The particular shifts are natural in the context of the array decomposition. "square" arrays and arrays where the column length is 3 yield now results for m-sequences. The array interpretation also determines the conditions under which the cross-correlation of GMW sequences is identical to the cross-correlation of the m-sequences used in the construction of the GMW pair, enabling known results for m-sequences to be applied. Then GMW sequence pairs with low cross-correlation can be selected.