{"title":"On the maximum size of variable-length non-overlapping codes","authors":"Geyang Wang, Qi Wang","doi":"10.1007/s10623-024-01445-3","DOIUrl":null,"url":null,"abstract":"<p>Non-overlapping codes are a set of codewords such that any nontrivial prefix of each codeword is not a nontrivial suffix of any codeword in the set, including itself. If the lengths of the codewords are variable, it is additionally required that every codeword is not contained in any other codeword as a subword. Let <i>C</i>(<i>n</i>, <i>q</i>) be the maximum size of a fixed-length non-overlapping code of length <i>n</i> over an alphabet of size <i>q</i>. The upper bound on <i>C</i>(<i>n</i>, <i>q</i>) has been well studied. However, the nontrivial upper bound on the maximum size of variable-length non-overlapping codes whose codewords have length at most <i>n</i> remains open. In this paper, by establishing a link between variable-length non-overlapping codes and fixed-length ones, we are able to show that the size of a <i>q</i>-ary variable-length non-overlapping code is upper bounded by <i>C</i>(<i>n</i>, <i>q</i>). Furthermore, we prove that the minimum average codeword length of a <i>q</i>-ary variable-length non-overlapping code with cardinality <span>\\(\\tilde{C}\\)</span>, is asymptotically no shorter than <span>\\(n-2\\)</span> as <i>q</i> approaches <span>\\(\\infty \\)</span>, where <i>n</i> is the smallest integer such that <span>\\(C(n-1, q) < \\tilde{C} \\le C(n,q)\\)</span>.</p>","PeriodicalId":11130,"journal":{"name":"Designs, Codes and Cryptography","volume":null,"pages":null},"PeriodicalIF":1.4000,"publicationDate":"2024-06-25","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/s10623-024-01445-3","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
Non-overlapping codes are a set of codewords such that any nontrivial prefix of each codeword is not a nontrivial suffix of any codeword in the set, including itself. If the lengths of the codewords are variable, it is additionally required that every codeword is not contained in any other codeword as a subword. Let C(n, q) be the maximum size of a fixed-length non-overlapping code of length n over an alphabet of size q. The upper bound on C(n, q) has been well studied. However, the nontrivial upper bound on the maximum size of variable-length non-overlapping codes whose codewords have length at most n remains open. In this paper, by establishing a link between variable-length non-overlapping codes and fixed-length ones, we are able to show that the size of a q-ary variable-length non-overlapping code is upper bounded by C(n, q). Furthermore, we prove that the minimum average codeword length of a q-ary variable-length non-overlapping code with cardinality \(\tilde{C}\), is asymptotically no shorter than \(n-2\) as q approaches \(\infty \), where n is the smallest integer such that \(C(n-1, q) < \tilde{C} \le C(n,q)\).
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.