{"title":"The R-transform as power map and its generalizations to higher degree","authors":"Alp Bassa , Ricardo Menares","doi":"10.1016/j.ffa.2024.102546","DOIUrl":"10.1016/j.ffa.2024.102546","url":null,"abstract":"<div><div>We give iterative constructions for irreducible polynomials over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> of degree <span><math><mi>n</mi><mo>⋅</mo><msup><mrow><mi>t</mi></mrow><mrow><mi>r</mi></mrow></msup></math></span> for all <span><math><mi>r</mi><mo>≥</mo><mn>0</mn></math></span>, starting from irreducible polynomials of degree <em>n</em>. The iterative constructions correspond modulo fractional linear transformations to compositions with power functions <span><math><msup><mrow><mi>x</mi></mrow><mrow><mi>t</mi></mrow></msup></math></span>. The <em>R</em>-transform introduced by Cohen is recovered as a particular case corresponding to <span><math><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>, hence we obtain a generalization of Cohen's <em>R</em>-transform (<span><math><mi>t</mi><mo>=</mo><mn>2</mn></math></span>) to arbitrary degrees <span><math><mi>t</mi><mo>≥</mo><mn>2</mn></math></span>. Important properties like self-reciprocity and invariance of roots under certain automorphisms are deduced from invariance under multiplication by appropriate roots of unity. Extending to quadratic extensions of <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> we recover and generalize a recursive construction of Panario, Reis and Wang.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"102 ","pages":"Article 102546"},"PeriodicalIF":1.2,"publicationDate":"2024-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142722321","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Webs and squabs of conics over finite fields","authors":"Nour Alnajjarine , Michel Lavrauw","doi":"10.1016/j.ffa.2024.102544","DOIUrl":"10.1016/j.ffa.2024.102544","url":null,"abstract":"<div><div>This paper is a contribution towards a solution for the longstanding open problem of classifying linear systems of conics over finite fields initiated by L. E. Dickson in 1908, through his study of the projective equivalence classes of pencils of conics in <span><math><mrow><mi>PG</mi></mrow><mo>(</mo><mn>2</mn><mo>,</mo><mi>q</mi><mo>)</mo></math></span>, for <em>q</em> odd. In this paper a set of complete invariants is determined for the projective equivalence classes of webs and of squabs of conics in <span><math><mrow><mi>PG</mi></mrow><mo>(</mo><mn>2</mn><mo>,</mo><mi>q</mi><mo>)</mo></math></span>, both for <em>q</em> odd and even. Our approach is mainly geometric, and involves a comprehensive study of the geometric and combinatorial properties of the Veronese surface in <span><math><mrow><mi>PG</mi></mrow><mo>(</mo><mn>5</mn><mo>,</mo><mi>q</mi><mo>)</mo></math></span>. The main contribution is the determination of the distribution of the different types of hyperplanes incident with the <em>K</em>-orbit representatives of points and lines of <span><math><mrow><mi>PG</mi></mrow><mo>(</mo><mn>5</mn><mo>,</mo><mi>q</mi><mo>)</mo></math></span>, where <span><math><mi>K</mi><mo>≅</mo><mrow><mi>PGL</mi></mrow><mo>(</mo><mn>3</mn><mo>,</mo><mi>q</mi><mo>)</mo></math></span>, is the subgroup of <span><math><mrow><mi>PGL</mi></mrow><mo>(</mo><mn>6</mn><mo>,</mo><mi>q</mi><mo>)</mo></math></span> stabilizing the Veronese surface.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"102 ","pages":"Article 102544"},"PeriodicalIF":1.2,"publicationDate":"2024-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142722322","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Some q-ary constacyclic BCH codes with length qm+12","authors":"Jin Li , Huilian Zhu , Shixin Zhu","doi":"10.1016/j.ffa.2024.102545","DOIUrl":"10.1016/j.ffa.2024.102545","url":null,"abstract":"<div><div>In this paper, we study some <em>q</em>-ary <em>λ</em>-constacyclic BCH codes of length <span><math><mfrac><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>+</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></math></span> with some large designed distances for <span><math><mrow><mi>ord</mi></mrow><mo>(</mo><mi>λ</mi><mo>)</mo><mo>=</mo><mn>2</mn></math></span> and <span><math><mn>2</mn><mo>+</mo><mo>⌈</mo><mfrac><mrow><mn>4</mn></mrow><mrow><mi>q</mi><mo>−</mo><mn>3</mn></mrow></mfrac><mo>⌉</mo><mo>≤</mo><mrow><mi>ord</mi></mrow><mo>(</mo><mi>λ</mi><mo>)</mo><mo>≤</mo><mi>q</mi><mo>−</mo><mn>1</mn></math></span> respectively, where <em>q</em> is an odd prime power and <span><math><mrow><mi>ord</mi></mrow><mo>(</mo><mi>λ</mi><mo>)</mo><mo>|</mo><mo>(</mo><mi>q</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span>. The dimensions and the lower bounds on the minimum distances of these codes are given by using recurrence relations and the introduced definitions of sequences. The code examples presented in this paper indicate that these codes have good parameters in general.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"102 ","pages":"Article 102545"},"PeriodicalIF":1.2,"publicationDate":"2024-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142707366","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Repeated-root constacyclic codes of length kslmpn over finite fields","authors":"Qi Zhang , Weiqiong Wang , Shuyu Luo , Yue Li","doi":"10.1016/j.ffa.2024.102542","DOIUrl":"10.1016/j.ffa.2024.102542","url":null,"abstract":"<div><div>For different odd primes <em>k</em>, <em>l</em>, <em>p</em>, and positive integers <em>s</em>, <em>m</em>, <em>n</em>, the polynomial <span><math><msup><mrow><mi>x</mi></mrow><mrow><msup><mrow><mi>k</mi></mrow><mrow><mi>s</mi></mrow></msup><msup><mrow><mi>l</mi></mrow><mrow><mi>m</mi></mrow></msup><msup><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></msup><mo>−</mo><mi>λ</mi></math></span> in <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mrow><mo>[</mo><mi>x</mi><mo>]</mo></mrow></math></span> is explicitly factorized, where <em>p</em> is the characteristic of <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>, <span><math><mi>λ</mi><mo>∈</mo><msubsup><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow><mrow><mo>⁎</mo></mrow></msubsup></math></span>. All repeated-root constacyclic codes and their dual codes of length <span><math><msup><mrow><mi>k</mi></mrow><mrow><mi>s</mi></mrow></msup><msup><mrow><mi>l</mi></mrow><mrow><mi>m</mi></mrow></msup><msup><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> are characterized. In addition, the characterization and enumeration of all linear complementary dual (LCD) cyclic and negacyclic codes of length <span><math><msup><mrow><mi>k</mi></mrow><mrow><mi>s</mi></mrow></msup><msup><mrow><mi>l</mi></mrow><mrow><mi>m</mi></mrow></msup><msup><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> are obtained.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"101 ","pages":"Article 102542"},"PeriodicalIF":1.2,"publicationDate":"2024-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142660203","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Complete description of measures corresponding to Abelian varieties over finite fields","authors":"Nikolai S. Nadirashvili , Michael A. Tsfasman","doi":"10.1016/j.ffa.2024.102543","DOIUrl":"10.1016/j.ffa.2024.102543","url":null,"abstract":"<div><div>We study probability measures corresponding to families of abelian varieties over a finite field. These measures play an important role in the Tsfasman–Vlăduţ theory of asymptotic zeta-functions defining completely the limit zeta-function of the family. J.-P. Serre, using results of R.M. Robinson on conjugate algebraic integers, described the possible set of measures than can correspond to families of abelian varieties over a finite field. The problem whether all such measures actually occur was left open. Moreover, Serre supposed that not all such measures correspond to abelian varieties (for example, the Lebesgue measure on a segment). Here we settle Serre's problem proving that Serre conditions are sufficient, and thus describe completely the set of measures corresponding to abelian varieties.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"101 ","pages":"Article 102543"},"PeriodicalIF":1.2,"publicationDate":"2024-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142660194","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Self-dual 2-quasi negacyclic codes over finite fields","authors":"Yun Fan, Yue Leng","doi":"10.1016/j.ffa.2024.102541","DOIUrl":"10.1016/j.ffa.2024.102541","url":null,"abstract":"<div><div>In this paper, we investigate the existence and asymptotic properties of self-dual 2-quasi negacyclic codes of length 2<em>n</em> over a finite field of cardinality <em>q</em>. When <em>n</em> is odd, we show that the <em>q</em>-ary self-dual 2-quasi negacyclic codes exist if and only if <span><math><mi>q</mi><mspace></mspace><mo>≢</mo><mo>−</mo><mn>1</mn><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><mn>4</mn><mo>)</mo></math></span>. When <em>n</em> is even, we prove that the <em>q</em>-ary self-dual 2-quasi negacyclic codes always exist. By using the technique introduced in this paper, we prove that <em>q</em>-ary self-dual 2-quasi negacyclic codes are asymptotically good.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"101 ","pages":"Article 102541"},"PeriodicalIF":1.2,"publicationDate":"2024-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142660258","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Intersecting families of polynomials over finite fields","authors":"Nika Salia , Dávid Tóth","doi":"10.1016/j.ffa.2024.102540","DOIUrl":"10.1016/j.ffa.2024.102540","url":null,"abstract":"<div><div>This paper demonstrates an analog of the Erdős–Ko–Rado theorem to polynomial rings over finite fields, affirmatively answering a conjecture of C. Tompkins.</div><div>A <em>k</em>-uniform family of subsets of a set of size <em>n</em> is <em>ℓ</em>-intersecting if any two subsets in the family intersect in at least <em>ℓ</em> elements. The study of such intersecting families is a core subject of extremal set theory, tracing its roots to the seminal 1961 Erdős–Ko–Rado theorem, which establishes a sharp upper bound on the size of these families. Here, we extend the Erdős–Ko–Rado theorem to polynomial rings over finite fields.</div><div>Specifically, we determine the largest possible size of a family of monic polynomials, each of degree <em>n</em>, over a finite field <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>, where every pair of polynomials in the family shares a common factor of degree at least <em>ℓ</em>. We prove that the upper bound for this size is <span><math><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi><mo>−</mo><mi>ℓ</mi></mrow></msup></math></span> and characterize all extremal families that achieve this maximum size.</div><div>Extending to triple-intersecting families, where every triplet of polynomials shares a common factor of degree at least <em>ℓ</em>, we prove that only trivial families achieve the corresponding upper bound. Moreover, by relaxing the conditions to include polynomials of degree at most <em>n</em>, we affirm that only trivial families achieve the corresponding upper bound.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"101 ","pages":"Article 102540"},"PeriodicalIF":1.2,"publicationDate":"2024-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142660192","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Partial difference sets with Denniston parameters in elementary abelian p-groups","authors":"Jingjun Bao , Qing Xiang , Meng Zhao","doi":"10.1016/j.ffa.2024.102539","DOIUrl":"10.1016/j.ffa.2024.102539","url":null,"abstract":"<div><div>Denniston <span><span>[12]</span></span> constructed partial difference sets (PDS) with parameters <span><math><mo>(</mo><msup><mrow><mn>2</mn></mrow><mrow><mn>3</mn><mi>m</mi></mrow></msup><mo>,</mo><mo>(</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi><mo>+</mo><mi>r</mi></mrow></msup><mo>−</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi></mrow></msup><mo>+</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>r</mi></mrow></msup><mo>)</mo><mo>(</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi></mrow></msup><mo>−</mo><mn>1</mn><mo>)</mo><mo>,</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi></mrow></msup><mo>−</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>r</mi></mrow></msup><mo>+</mo><mo>(</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi><mo>+</mo><mi>r</mi></mrow></msup><mo>−</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi></mrow></msup><mo>+</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>r</mi></mrow></msup><mo>)</mo><mo>(</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>r</mi></mrow></msup><mo>−</mo><mn>2</mn><mo>)</mo><mo>,</mo><mo>(</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi><mo>+</mo><mi>r</mi></mrow></msup><mo>−</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi></mrow></msup><mo>+</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>r</mi></mrow></msup><mo>)</mo><mo>(</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>r</mi></mrow></msup><mo>−</mo><mn>1</mn><mo>)</mo><mo>)</mo></math></span> in elementary abelian groups of order <span><math><msup><mrow><mn>2</mn></mrow><mrow><mn>3</mn><mi>m</mi></mrow></msup></math></span> for all <span><math><mi>m</mi><mo>≥</mo><mn>2</mn></math></span> and <span><math><mn>1</mn><mo>≤</mo><mi>r</mi><mo><</mo><mi>m</mi></math></span>. These PDS arise from maximal arcs in the Desarguesian projective planes PG<span><math><mo>(</mo><mn>2</mn><mo>,</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi></mrow></msup><mo>)</mo></math></span>. Davis et al. <span><span>[10]</span></span> and also De Winter <span><span>[13]</span></span> presented constructions of PDS with Denniston parameters <span><math><mo>(</mo><msup><mrow><mi>p</mi></mrow><mrow><mn>3</mn><mi>m</mi></mrow></msup><mo>,</mo><mo>(</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>m</mi><mo>+</mo><mi>r</mi></mrow></msup><mo>−</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>+</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>r</mi></mrow></msup><mo>)</mo><mo>(</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>−</mo><mn>1</mn><mo>)</mo><mo>,</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>−</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>r</mi></mrow></msup><mo>+</mo><mo>(</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>m</mi><mo>+</mo><mi>r</mi></mrow></msup><mo>−</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>+</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>r</mi></mrow></msup><mo>)</mo><mo>(</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>r</mi></mrow></msup><mo>−</mo><mn>2</mn><mo>)</mo><mo>,</mo><mo>(</mo><msup><mrow><mi>p</mi>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"101 ","pages":"Article 102539"},"PeriodicalIF":1.2,"publicationDate":"2024-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142660193","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Asymptotic distributions of the number of zeros of random polynomials in Hayes equivalence class over a finite field","authors":"Zhicheng Gao","doi":"10.1016/j.ffa.2024.102524","DOIUrl":"10.1016/j.ffa.2024.102524","url":null,"abstract":"<div><div>Hayes equivalence is defined on monic polynomials over a finite field <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> in terms of the prescribed leading coefficients and the residue classes modulo a given monic polynomial <em>Q</em>. We study the distribution of the number of zeros in a random polynomial over finite fields in a given Hayes equivalence class. It is well known that the number of distinct zeros of a random polynomial over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> is asymptotically Poisson with mean 1. We show that this is also true for random polynomials in any given Hayes equivalence class. Asymptotic formulas are also given for the number of such polynomials when the degree of such polynomials is proportional to <em>q</em> and the degree of <em>Q</em> and the number of prescribed leading coefficients are bounded by <span><math><msqrt><mrow><mi>q</mi></mrow></msqrt></math></span>. When <span><math><mi>Q</mi><mo>=</mo><mn>1</mn></math></span>, the problem is equivalent to the study of the distance distribution in Reed-Solomon codes. Our asymptotic formulas extend some earlier results and imply that all words for a large family of Reed-Solomon codes are ordinary, which further supports the well-known <em>Deep-Hole</em> Conjecture.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"101 ","pages":"Article 102524"},"PeriodicalIF":1.2,"publicationDate":"2024-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142594089","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Quasi-polycyclic and skew quasi-polycyclic codes over Fq","authors":"Tushar Bag, Daniel Panario","doi":"10.1016/j.ffa.2024.102536","DOIUrl":"10.1016/j.ffa.2024.102536","url":null,"abstract":"<div><div>In this research, our focus is on investigating 1-generator right quasi-polycyclic (QPC) codes over fields. We provide a detailed description of how linear codes with substantial minimum distances can be constructed from QPC codes. We analyze dual QPC codes under various inner products and use them to construct quantum error-correcting codes. Furthermore, our research includes a dedicated section that delves into the area of skew quasi-polycyclic (SQPC) codes, investigating their properties and the role of generators in their construction. This section expands our study to encompass the intriguing area of SQPC codes, offering insights into the non-commutative version of QPC codes, their characteristics and generator structures. Our work deals with the structural properties of QPC, skew polycyclic and SQPC codes, shedding light on their potential for enhancing the field of coding theory.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"101 ","pages":"Article 102536"},"PeriodicalIF":1.2,"publicationDate":"2024-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142561480","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}