{"title":"On prefer-one sequences","authors":"Yupeng Jiang, Ming Li, Ying Gao, Dongdai Lin","doi":"10.1007/s10623-024-01425-7","DOIUrl":null,"url":null,"abstract":"<p>We study the sequences generated by prefer-one rule with different initial vectors. Firstly, we give upper bounds of their periods and for initial vectors with Hamming weight one, we prove that the generated sequences are modified de Bruijn sequences. Moreover, for two of them, we give the truth tables of their feedback functions. We also investigate the feedback functions of prefer-one de Bruijn sequences. For order <i>n</i> prefer-one de Bruijn sequence, we give linear and quadratic terms in its feedback function and prove that the number of degree <span>\\(n-2\\)</span> terms has the same parity as <i>n</i>. The statistical result for small <i>n</i> shows that about half of all terms occur in the feedback functions.</p>","PeriodicalId":11130,"journal":{"name":"Designs, Codes and Cryptography","volume":"40 1","pages":""},"PeriodicalIF":1.4000,"publicationDate":"2024-06-10","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/s10623-024-01425-7","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}
引用次数: 0
Abstract
We study the sequences generated by prefer-one rule with different initial vectors. Firstly, we give upper bounds of their periods and for initial vectors with Hamming weight one, we prove that the generated sequences are modified de Bruijn sequences. Moreover, for two of them, we give the truth tables of their feedback functions. We also investigate the feedback functions of prefer-one de Bruijn sequences. For order n prefer-one de Bruijn sequence, we give linear and quadratic terms in its feedback function and prove that the number of degree \(n-2\) terms has the same parity as n. The statistical result for small n shows that about half of all terms occur in the feedback functions.
我们研究了用 prefer-one 规则生成的具有不同初始向量的序列。首先，我们给出了它们的周期上限，对于汉明权重为 1 的初始向量，我们证明了生成的序列是修正的德布鲁因序列。此外，我们还给出了其中两个序列的反馈函数真值表。我们还研究了优选一德布鲁因序列的反馈函数。对于 n 阶 prefer-one de Bruijn 序列，我们给出了其反馈函数中的线性项和二次项，并证明了度（n-2）项的数量与 n 具有相同的奇偶性。
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.
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