Barker sequences of odd length

A Barker sequence is a binary sequence for which all nontrivial aperiodic autocorrelations are at most 1 in magnitude. An old conjecture due to Turyn asserts that there is no Barker sequence of length greater than 13. In 1961, Turyn and Storer gave an elementary, though somewhat complicated, proof that this conjecture holds for odd lengths. We give a new and simpler proof of this result.Comment: 6 pages, this note supersedes the main result of arXiv:1409.143

Hadamard matrices modulo 5

In this paper we introduce modular symmetric designs and use them to study the existence of Hadamard matrices modulo 5. We prove that there exist 5-modular Hadamard matrices of order n if and only if n != 3, 7 (mod 10) or n != 6, 11. In particular, this solves the 5-modular version of the Hadamard conjecture.Comment: 7 pages, submitted to JC

The merit factor of binary arrays derived from the quadratic character

We calculate the asymptotic merit factor, under all cyclic rotations of rows and columns, of two families of binary two-dimensional arrays derived from the quadratic character. The arrays in these families have size p x q, where p and q are not necessarily distinct odd primes, and can be considered as two-dimensional generalisations of a Legendre sequence. The asymptotic values of the merit factor of the two families are generally different, although the maximum asymptotic merit factor, taken over all cyclic rotations of rows and columns, equals 36/13 for both families. These are the first non-trivial theoretical results for the asymptotic merit factor of families of truly two-dimensional binary arrays.Comment: minor correction

Выбор зондирующего сигнала для радиолокационного комплекса в режиме синтезированной апертуры антенны

В работе рассмотрены методы формирования зондирующего сигнала для радиолокационной станции в режиме синтезированной апертуры антенны, обеспечивающие минимальный уровень боковых лепестков в плоскости дальность-азимут. Показано, что наилучшим по указанным критериям является фазоманипулированный сигнал с минимально возможным уровнем боковых лепестков импульсной автокорреляционной функции.Работа выполнена при финансовой поддержке в рамках государственных контрактов № 02.G25.31.0204, №2.9140.2017/БЧ, №2.2226.2017/ПЧ и гранта РФФИ №15-07-99514а

Under a mild condition, Ryser\u27s Conjecture holds for every ( n:= 4h^2) with h>1 odd and non square-free

We prove, under a mild condition, that there is no circulant Hadamard matrix ( H) with (n >4) rows when (sqrt{n/4}) is not square-free. The proof introduces a new method to attack Ryser\u27s Conjecture, that is a long standing difficult conjecture