Advances in the merit factor problem for binary sequences

Abstract

The identification of binary sequences with large merit factor (small mean-squared aperiodic autocorrelation) is an old problem of complex analysis and combinatorial optimization, with practical importance in digital communications engineering and condensed matter physics. We establish the asymptotic merit factor of several families of binary sequences and thereby prove various conjectures, explain numerical evidence presented by other authors, and bring together within a single framework results previously appearing in scattered form. We exhibit, for the first time, families of skew-symmetric sequences whose asymptotic merit factor is as large as the best known value (an algebraic number greater than 6.34) for all binary sequences; this is interesting in light of Golay's conjecture that the subclass of skew-symmetric sequences has asymptotically optimal merit factor. Our methods combine Fourier analysis, estimation of character sums, and estimation of the number of lattice points in polyhedra.

Lq Norms of Fekete and Related Polynomials

“…However every such polynomial is again a Galois polynomial. It should also be noted that our methods can be used to establish similar results for polynomials obtained by periodically appending or truncating monomials in Fekete or Galois polynomials, as considered in [22] and [21].…”

“…Fekete polynomials appear frequently in the context of extremal polynomial problems [30], [19], [23], [9], [5], [4], [22], [21], [24] and have been studied extensively now for over a century [15]. Erdélyi [13] established the order of growth of the L α norm of Fekete polynomials.…”

Sequences with Low Correlation

“…In these respective cases, we say that this new sequence g is f truncated to m/ℓ times its usual length or f appended to m/ℓ times its usual length. Jedwab, Katz, and Schmidt [19,Theorem 2.2] proved that if one applies this procedure to m-sequences, one can produce families with an asymptotic autocorrelation demerit factor of 0.299 . .…”

“…of Rudin-Shapiro sequences has asymptotic demerit factor 1/3. [4,Theorem 1] showed that the companion sequences in the construction (19) were related to the main sequences by g n (z) = (−1) n z 2 n −1 f n (−1/z) for every n. For any polynomial h(z) ∈ C[z], we define the reciprocal polynomial of h(z), denoted h * (z), to be z deg h h(1/z), that is, the polynomial obtained from h by writing the coefficients in reverse order. Then the result of Brillhart and Carlitz becomes g n (z) = (−1) 2 n +n−1 f * n (−z), so that we could restate the construction without the companion sequences:…”

Low Correlation Sequences From Linear Combinations of Characters

“…This is achieved by sequences derived from finite field characters, specifically quadratic characters (also known as Legendre symbols). See [14,Theorem 1.1], [19,Theorem 1.5], and [13] for details. Other sequences with good correlation properties derived from finite field characters include the maximal linear recursive sequences (msequences), which are used extensively in radar and communications networks.…”