{"title":"二进制序列自相关扣分因子的矩","authors":"Daniel J. Katz, Miriam E. Ramirez","doi":"10.1007/s10623-024-01482-y","DOIUrl":null,"url":null,"abstract":"<p>Sequences with low aperiodic autocorrelation are used in communications and remote sensing for synchronization and ranging. The autocorrelation demerit factor of a sequence is the sum of the squared magnitudes of its autocorrelation values at every nonzero shift when we normalize the sequence to have unit Euclidean length. The merit factor, introduced by Golay, is the reciprocal of the demerit factor. We consider the uniform probability measure on the <span>\\(2^\\ell \\)</span> binary sequences of length <span>\\(\\ell \\)</span> and investigate the distribution of the demerit factors of these sequences. Sarwate and Jedwab have respectively calculated the mean and variance of this distribution. We develop new combinatorial techniques to calculate the <i>p</i>th central moment of the demerit factor for binary sequences of length <span>\\(\\ell \\)</span>. These techniques prove that for <span>\\(p\\ge 2\\)</span> and <span>\\(\\ell \\ge 4\\)</span>, all the central moments are strictly positive. For any given <i>p</i>, one may use the technique to obtain an exact formula for the <i>p</i>th central moment of the demerit factor as a function of the length <span>\\(\\ell \\)</span>. Jedwab’s formula for variance is confirmed by our technique with a short calculation, and we go beyond previous results by also deriving an exact formula for the skewness. A computer-assisted application of our method also obtains exact formulas for the kurtosis, which we report here, as well as the fifth central moment.</p>","PeriodicalId":11130,"journal":{"name":"Designs, Codes and Cryptography","volume":"26 1","pages":""},"PeriodicalIF":1.4000,"publicationDate":"2024-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Moments of autocorrelation demerit factors of binary sequences\",\"authors\":\"Daniel J. Katz, Miriam E. Ramirez\",\"doi\":\"10.1007/s10623-024-01482-y\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Sequences with low aperiodic autocorrelation are used in communications and remote sensing for synchronization and ranging. The autocorrelation demerit factor of a sequence is the sum of the squared magnitudes of its autocorrelation values at every nonzero shift when we normalize the sequence to have unit Euclidean length. The merit factor, introduced by Golay, is the reciprocal of the demerit factor. We consider the uniform probability measure on the <span>\\\\(2^\\\\ell \\\\)</span> binary sequences of length <span>\\\\(\\\\ell \\\\)</span> and investigate the distribution of the demerit factors of these sequences. Sarwate and Jedwab have respectively calculated the mean and variance of this distribution. We develop new combinatorial techniques to calculate the <i>p</i>th central moment of the demerit factor for binary sequences of length <span>\\\\(\\\\ell \\\\)</span>. These techniques prove that for <span>\\\\(p\\\\ge 2\\\\)</span> and <span>\\\\(\\\\ell \\\\ge 4\\\\)</span>, all the central moments are strictly positive. For any given <i>p</i>, one may use the technique to obtain an exact formula for the <i>p</i>th central moment of the demerit factor as a function of the length <span>\\\\(\\\\ell \\\\)</span>. Jedwab’s formula for variance is confirmed by our technique with a short calculation, and we go beyond previous results by also deriving an exact formula for the skewness. A computer-assisted application of our method also obtains exact formulas for the kurtosis, which we report here, as well as the fifth central moment.</p>\",\"PeriodicalId\":11130,\"journal\":{\"name\":\"Designs, Codes and Cryptography\",\"volume\":\"26 1\",\"pages\":\"\"},\"PeriodicalIF\":1.4000,\"publicationDate\":\"2024-09-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Designs, Codes and Cryptography\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s10623-024-01482-y\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"COMPUTER SCIENCE, THEORY & METHODS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Designs, Codes and Cryptography","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10623-024-01482-y","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}
Moments of autocorrelation demerit factors of binary sequences
Sequences with low aperiodic autocorrelation are used in communications and remote sensing for synchronization and ranging. The autocorrelation demerit factor of a sequence is the sum of the squared magnitudes of its autocorrelation values at every nonzero shift when we normalize the sequence to have unit Euclidean length. The merit factor, introduced by Golay, is the reciprocal of the demerit factor. We consider the uniform probability measure on the \(2^\ell \) binary sequences of length \(\ell \) and investigate the distribution of the demerit factors of these sequences. Sarwate and Jedwab have respectively calculated the mean and variance of this distribution. We develop new combinatorial techniques to calculate the pth central moment of the demerit factor for binary sequences of length \(\ell \). These techniques prove that for \(p\ge 2\) and \(\ell \ge 4\), all the central moments are strictly positive. For any given p, one may use the technique to obtain an exact formula for the pth central moment of the demerit factor as a function of the length \(\ell \). Jedwab’s formula for variance is confirmed by our technique with a short calculation, and we go beyond previous results by also deriving an exact formula for the skewness. A computer-assisted application of our method also obtains exact formulas for the kurtosis, which we report here, as well as the fifth central moment.
期刊介绍:
Designs, Codes and Cryptography is an archival peer-reviewed technical journal publishing original research papers in the designated areas. There is a great deal of activity in design theory, coding theory and cryptography, including a substantial amount of research which brings together more than one of the subjects. While many journals exist for each of the individual areas, few encourage the interaction of the disciplines.
The journal was founded to meet the needs of mathematicians, engineers and computer scientists working in these areas, whose interests extend beyond the bounds of any one of the individual disciplines. The journal provides a forum for high quality research in its three areas, with papers touching more than one of the areas especially welcome.
The journal also considers high quality submissions in the closely related areas of finite fields and finite geometries, which provide important tools for both the construction and the actual application of designs, codes and cryptographic systems. In particular, it includes (mostly theoretical) papers on computational aspects of finite fields. It also considers topics in sequence design, which frequently admit equivalent formulations in the journal’s main areas.
Designs, Codes and Cryptography is mathematically oriented, emphasizing the algebraic and geometric aspects of the areas it covers. The journal considers high quality papers of both a theoretical and a practical nature, provided they contain a substantial amount of mathematics.