{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:56:11Z","timestamp":1760144171636,"version":"build-2065373602"},"reference-count":30,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2024,3,27]],"date-time":"2024-03-27T00:00:00Z","timestamp":1711497600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>It has been known since the 1970\u2019s that the difference of the non-zero weights of a projective Fq-linear two-weight code has to be a power of the characteristic of the underlying field. Here, we study non-projective two-weight codes and, e.g., show the same result under mild extra conditions. For small dimensions we give exhaustive enumerations of the feasible parameters in the binary case.<\/jats:p>","DOI":"10.3390\/e26040289","type":"journal-article","created":{"date-parts":[[2024,3,27]],"date-time":"2024-03-27T13:39:56Z","timestamp":1711546796000},"page":"289","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Non-Projective Two-Weight Codes"],"prefix":"10.3390","volume":"26","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4597-2041","authenticated-orcid":false,"given":"Sascha","family":"Kurz","sequence":"first","affiliation":[{"name":"Mathematisches Institut, Universit\u00e4t Bayreuth, D-95440 Bayreuth, Germany"},{"name":"Department of Data Science, Friedrich-Alexander-Universit\u00e4\u00e4t Erlangen-N\u00fcrnberg, D-91058 Erlangen, Germany"}]}],"member":"1968","published-online":{"date-parts":[[2024,3,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"47","DOI":"10.1016\/0012-365X(72)90024-6","article-title":"Weights of linear codes and strongly regular normed spaces","volume":"3","author":"Delsarte","year":"1972","journal-title":"Discret. Math."},{"key":"ref_2","unstructured":"Brouwer, A.E. (2021). Concise Encyclopedia of Coding Theory, Chapman and Hall\/CRC."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"2262","DOI":"10.1016\/j.disc.2017.04.016","article-title":"A construction of linear codes and strongly regular graphs from q-polynomials","volume":"340","author":"Luo","year":"2017","journal-title":"Discret. Math."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"112318","DOI":"10.1016\/j.disc.2021.112318","article-title":"On two-weight codes","volume":"344","author":"Boyvalenkov","year":"2021","journal-title":"Discret. Math."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1112\/blms\/18.2.97","article-title":"The geometry of two-weight codes","volume":"18","author":"Calderbank","year":"1986","journal-title":"Bull. Lond. Math. Soc."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Brouwer, A.E., and Maldeghem, H.V. (2022). Strongly Regular Graphs, Cambridge University Press.","DOI":"10.1017\/9781009057226"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1993","DOI":"10.1007\/s10623-021-00896-2","article-title":"A family of projective two-weight linear codes","volume":"89","author":"Heng","year":"2021","journal-title":"Des. Codes Cryptogr."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1451","DOI":"10.1016\/j.dam.2007.03.006","article-title":"Constructing two-weight codes with prescribed groups of automorphisms","volume":"155","author":"Kohnert","year":"2007","journal-title":"Discret. Appl. Math."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"202","DOI":"10.1016\/j.disc.2014.11.007","article-title":"Geometric constructions of two-character sets","volume":"338","author":"Pavese","year":"2015","journal-title":"Discret. Math."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"102186","DOI":"10.1016\/j.ffa.2023.102186","article-title":"Two new classes of projective two-weight linear codes","volume":"88","author":"Zhu","year":"2023","journal-title":"Finite Fields Their Appl."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"59","DOI":"10.1007\/s10623-006-0019-1","article-title":"Projective two-weight codes with small parameters and their corresponding graphs","volume":"41","author":"Bouyukliev","year":"2006","journal-title":"Des. Codes Cryptogr."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"261","DOI":"10.1023\/A:1008294128110","article-title":"The correspondence between projective codes and 2-weight codes","volume":"11","author":"Brouwer","year":"1997","journal-title":"Des. Codes Cryptogr."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"328","DOI":"10.3934\/amc.2023032","article-title":"Dual transform and projective self-dual codes","volume":"18","author":"Bouyukliev","year":"2024","journal-title":"Adv. Math. Commun."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"4049","DOI":"10.1016\/j.disc.2008.12.002","article-title":"Classification of Griesmer codes and dual transform","volume":"309","author":"Bouyukliev","year":"2009","journal-title":"Discret. Math."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"842","DOI":"10.1016\/j.disc.2007.07.044","article-title":"On optimal non-projective ternary linear codes","volume":"308","author":"Takenaka","year":"2008","journal-title":"Discret. Math."},{"key":"ref_16","unstructured":"Bouyuklieva, S., and Bouyukliev, I. (2017, January 7\u20138). Dual transform through characteristic vectors. Proceedings of the International Workshop OCRT, Sofia, Bulgaria."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"372","DOI":"10.1016\/j.ffa.2017.12.010","article-title":"The classification of antipodal two-weight linear codes","volume":"50","author":"Jungnickel","year":"2018","journal-title":"Finite Fields Their Appl."},{"key":"ref_18","unstructured":"Vega, G., and V\u00e1zquez, C.A. (2012). Arithmetic of Finite Fields, Proceedings of the 4th International Workshop, WAIFI 2012, Bochum, Germany, 16\u201319 July 2012, Springer. Proceedings 4."},{"key":"ref_19","unstructured":"Duc, T.D. (2019). Non-projective cyclic codes whose check polynomial contains two zeros. arXiv."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1023\/A:1021823703405","article-title":"On a particular class of minihypers and its applications. I. The result for general q","volume":"28","author":"Govaerts","year":"2003","journal-title":"Des. Codes Cryptogr."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s10623-007-9136-8","article-title":"Ring geometries, two-weight codes, and strongly regular graphs","volume":"48","author":"Byrne","year":"2008","journal-title":"Des. Codes Cryptogr."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"R37","DOI":"10.37236\/1375","article-title":"Codes and projective multisets","volume":"5","author":"Dodunekov","year":"1998","journal-title":"Electron. J. Comb."},{"key":"ref_23","first-page":"181","article-title":"Every equidistant linear code is a sequence of dual Hamming codes","volume":"Volume 18","author":"Bonisoli","year":"1984","journal-title":"ARS Combinatoria"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Greferath, M., Pav\u010devi\u0107, M.O., Silberstein, N., and V\u00e1zquez-Castro, M.\u00c1. (2018). Network Coding and Subspace Designs, Springer.","DOI":"10.1007\/978-3-319-70293-3"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1023\/A:1008302520762","article-title":"An introduction to divisible codes","volume":"17","author":"Ward","year":"1999","journal-title":"Des. Codes Cryptogr."},{"key":"ref_26","unstructured":"Heinlein, D., Honold, T., Kiermaier, M., Kurz, S., and Wassermann, A. (20017, January 18\u201322). Projective divisible binary codes. Proceedings of the 10th International Workshop on Coding and Cryptography, Saint Petersburg, Russia."},{"key":"ref_27","unstructured":"Kurz, S. (2021). Divisible codes. arXiv."},{"key":"ref_28","unstructured":"Betten, A., Braun, M., Fripertinger, H., Kerber, A., Kohnert, A., and Wassermann, A. (2006). Error-Correcting Linear Codes: Classification by Isometry and Applications, Springer Science & Business Media."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1693","DOI":"10.1016\/j.dam.2006.03.004","article-title":"On the binary projective codes with dimension 6","volume":"154","author":"Bouyukliev","year":"2006","journal-title":"Discret. Appl. Math."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"7807","DOI":"10.1109\/TIT.2021.3114280","article-title":"Computer classification of linear codes","volume":"67","author":"Bouyukliev","year":"2021","journal-title":"IEEE Trans. Inf. 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