LABS problem and ground state spin glasses system

Abstract

Abstract.In our work we demonstrate the new results of an exhaustive search for optimal binary sequences with minimum peak sidelobe (MPS) up to length N=85. The design problem for law autocorrelation binary sequences (LABS) is a notoriously difficult computational problem which is numbered as the problem number 005 in CSPLib. In statistical physics LABS problem can be interrepted as the energy of N iteracting Ising spins. This is a Bernasconi model. Due to this connection to physics we refer a binary sequence as one-dimensional spin lattice. At this assumption optimal binary sequences by merit factor (MF) criteria are the ground-state spin system without disorder which exhibits a glassy regime.Design method for ground-state level quantum system for one-dimensional Ising spin chain [1] consisting from N mutual equal-distance particles is offered. Within the frame of Bernasconi model [2] the design problem of simplest quantum system presented like onedimensional Ising model is reduced to the design problem of binary sequences with lowest level of energy of sidelobes [3,4] of aperiodic autocorrelation. In fact, the Bernasconi model exhibits features of a glass transition like a jump in the specific heat and slow dynammics and aging.The problem of the glass state remains one major unsolved issued in condensed matter theory. Despite an enormous body of experimental and numerical data and quite detailed phenomenological theories, there is no fully stisfactory microsopic model for the glass state. Within the Bernascni model the high temperature phase of Ising spin system reproduces exactly an approximation due to Golay [3]. For the low-temperature regime, analytical results are rare -especialy for the ground state are not known. Low temperature glass phase is rather similar to the low temperature phase of Derrida's random energy model [15].There are two criteria for the optimality of binary sequences with low levels of aperiodic autocorrelation: minmum peak sidlobe (PSL) and maximum MF. The effective computing methods for an exhaustive search of binary sequences with maximum MF are presented in [5][6][7]. Optimal binary sequences by MF criteria are consructed for the lengths N= [2,66]. Enumerative algorihms (complete or partial) are limited to small values of N<300 by the eponential complexity of local search algorithms. Analytical binary sequences for unlimited N with current record of an symptotic merit factor of MF=6.340261 is set called appended rotated Legenfre sequences.