1 In Searching of Long Skew-symmetric Binary Sequences with High Merit Factors Janez Brest, Senior Member, IEEE, and Borko Boˇ skovi´ c, Member, IEEE Abstract—In this paper we present best-known merit fac- tors of longer binary sequences with odd length. Finding low autocorrelation binary sequences with optimal merit factors is difficult optimization problem. High performance computations with execution of a stochastic algorithm in parallel, enable us searching skew-symmetric binary sequences with high merit factors. After experimental work, as results we present sequences with odd length between 301 and 401 that are skew-symmetric and have merit factor greater than 7. Moreover, now all sequences with odd length between 301 and 401 with > 7 have been found. Index Terms—Golay’s merit factor, binary sequences, aperiodic autocorrelation, best-known results. I. I NTRODUCTION Low autocorrelation binary sequences (LABS) play impor- tant role in communication and radar applications, due to low aperiodic autocorrelation property [32]. Finding LABS with optimal merit factors is a challenging optimization problem. Nowadays, a parallel computation can be applied to tackle hard optimization problems. A power of several computers that are not necessary placed in the same location, but they can be spread also oversee is joined together in solving real-world problems. A grid computing is used to make computations for finding (binary) sequences in [27, 22, 7, 8]. Binary sequences with low autocorrelation function prop- erties are important in many areas, such as communication engineering [33, 35, 34] and in statistical mechanics [25, 3, 23]. Also in mathematics, this problem (see Littlewood polynomial) has attracted sustained interest [14, 17]. Long binary sequences are essential for various applications of the coded exposure process [20, 21]. A binary sequence = 1 2 ... has all entries either +1 or −1. Here, denotes a length the sequence. The autocorrelations of are defined ( ) = − ∑ =1 + , for = 0, 1,..., − 1, (1) and the energy of is ( ) = −1 ∑ =1 2 ( ) . (2) The authors acknowledge the financial support from the Slovenian Research Agency (research core funding No. P2-0041 – Computer Systems, Method- ologies, and Intelligent Services). The authors would also like to acknowledge the Slovenian Initiative for National Grid (SLING) for using a computation power for performing the experiments in this paper. J. Brest and B. Boˇ skovi´ c are with the Institute of Computer Sci- ence, Faculty of Electrical Engineering and Computer Science, Uni- versity of Maribor, Koroˇ ska c. 46, 2000 Maribor, Slovenia (e-mail: {janez.brest, borko.boskovic}@um.si). Note, that ( ) is defined as the sum of the squares of all off-peak autocorrelations. The LABS problem involves assigning values to the that minimize ( ) or maximize the merit factor ( ) [12]: ( ) = 2 2 ( ) . (3) The merit factor is a measure of the quality of the sequence in terms of engineering applications [6]. The skew-symmetric sequences have odd length with = 2 − 1 and satisfy + = (−1) − , = 1, 2,..., − 1. (4) which implies that = 0 for all odd . The restriction of the problem to skew-symmetric sequences reduces the sequence’s effective length from to approximately /2. Which means, the dimension of the problem and the search space are reduced. The search space is reduced from 2 to approximately 2 ( /2) [25]. Note that optimal skew-symmetric solutions might not be optimal for the whole search space. Besides the merit factor, another metric for LABS problem is the Peak Sidelobe Level ( ( ) = max −1 =1 | ( )|) [19]. A sequence with the optimal PSL usually has a merit factor which is much lower than the optimal merit factor, and vice versa. In this paper, our key focus is to search for long sequences with high merit factors. One of the main challenges when solving the LABS prob- lem using the incomplete search is how to implement a calculation of energy efficiently and researchers developed an efficient implementation of the energy calculation [11, 15, 7, 24]. Note that similar efficient calculation can be applied to finding a skew-symmetric solution of the odd length problem instances. So far, a few binary sequences longer than 300 with a merit factor above 7 are known. These sequences are: 301, 303, 304 and 381 [7, 4], and they have merit factors of 7.4827, 7.2462, 7.035, and 7.0893, respectively. In this paper, we used the xLastovka [8] stochastic algorithm for searching skew-symmetric binary sequences. The obtained skew-symmetric solutions might not be optimal for each LABS problem instance. However, the paper’s main goal is not to build yet another heuristic optimization algorithm but to shed some light on the searching binary sequences longer than 300. The paper will focus on the quality of the obtained results and good merit factors. In particular, we investigate through extensive experiment runs the influence of dimensionality of sequences. At the end of this research we succeeded to find a number of binary sequences that have merit factors greater arXiv:2011.00068v1 [cs.IT] 30 Oct 2020