{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,6]],"date-time":"2026-09-06T18:18:57Z","timestamp":1788718737431,"version":"build-2803163510"},"reference-count":46,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","funder":[{"DOI":"10.13039\/501100001691","name":"Japan Society for the Promotion of Science","doi-asserted-by":"publisher","award":["15H03636"],"award-info":[{"award-number":["15H03636"]}],"id":[{"id":"10.13039\/501100001691","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001691","name":"Japan Society for the Promotion of Science","doi-asserted-by":"publisher","award":["18H01133"],"award-info":[{"award-number":["18H01133"]}],"id":[{"id":"10.13039\/501100001691","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001691","name":"Japan Society for the Promotion of Science","doi-asserted-by":"publisher","award":["19K14585"],"award-info":[{"award-number":["19K14585"]}],"id":[{"id":"10.13039\/501100001691","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Discrete Math."],"published-print":{"date-parts":[[2021,1]]},"abstract":"<jats:p>BCH codes are among the best practical cyclic codes widely used in consumer electronics, communication systems, and storage devices. However, not much is known about BCH codes with large minimum distance. In this paper, we consider narrow-sense BCH codes of length $n = \\frac{q^m-1}{N}$ with designed distance $\\delta = \\frac{s}{q-1}n$ proportional to $n$, where $N$ divides $\\frac{q^m-1}{q-1}$ and $1 \\le s \\le q-1$. We determine both their dimensions and minimum distances. In particular, when $N=1$, the codes are primitive, with minimum distance $d=\\frac{s}{q-1}(q^m-1)$ and dimension $k = (q-s)^m$. The general result on code dimensions is achieved by applying generating functions and inverse discrete Fourier transforms to an enumeration problem.<\/jats:p>","DOI":"10.1137\/19m1260876","type":"journal-article","created":{"date-parts":[[2021,2,11]],"date-time":"2021-02-11T10:38:52Z","timestamp":1613039932000},"page":"179-193","source":"Crossref","is-referenced-by-count":29,"title":["BCH Codes with Minimum Distance Proportional to Code Length"],"prefix":"10.1137","volume":"35","author":[{"given":"Satoshi","family":"Noguchi","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xiao-Nan","family":"Lu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Masakazu","family":"Jimbo","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ying","family":"Miao","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2021,2,11]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1109\/TIT.2006.890730"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1109\/18.135638"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1109\/18.272483"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1002\/j.1538-7305.1967.tb03175.x"},{"key":"atypb5","unstructured":"E. R. Berlekamp,\n                      Algebraic Coding Theory\n                      , McGraw-Hill, New York, 1968."},{"key":"atypb6","unstructured":"A. Betten, M. Braun, H. Fripertinger, A. Kerber, A. Kohnert, and A. Wassermann,\n                      Error-Correcting Linear Codes: Classification by Isometry and Applications\n                      , Springer-Verlag, Berlin, 2006."},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1016\/S0019-9958(60)90870-6"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1016\/S0019-9958(60)90287-4"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1109\/18.50397"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1109\/18.333859"},{"key":"atypb11","unstructured":"P. Charpin,\n                      Open Problems on Cyclic Codes\n                      , in Handbook of Coding Theory, Vol. 1, V. S. Pless and W. C. Huffman, eds., North-Holland, Amsterdam, 1998, pp. 963-1063."},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1109\/18.605611"},{"key":"atypb13","doi-asserted-by":"crossref","unstructured":"C. Ding,\n                      Codes from Difference Sets\n                      , World Scientific, Singapore, 2015.","DOI":"10.1142\/9283"},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.1109\/TIT.2015.2470251"},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.1109\/TIT.2015.2409838"},{"key":"atypb16","doi-asserted-by":"publisher","DOI":"10.1016\/j.ffa.2016.12.009"},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.1137\/0109020"},{"key":"atypb18","unstructured":"M. Grassl,\n                      Bounds on the Minimum Distance of Linear Codes and Quantum Codes\n                      , available online athttp:\/\/www.codetables.de, 2007 (accessed on 2020-10-06)."},{"key":"atypb19","first-page":"57","volume":"32","author":"Guenda K.","year":"2008","journal-title":"Ann. Sci. Math. Qu\u00e9bec"},{"key":"atypb20","first-page":"147","volume":"2","author":"Hocquenghem A.","year":"1959","journal-title":"Chiffres"},{"key":"atypb21","doi-asserted-by":"crossref","unstructured":"W. C. Huffman and V. Pless,\n                      Fundamentals of Error-Correcting Codes\n                      , Cambridge University Press, Cambridge, UK, 2003.","DOI":"10.1017\/CBO9780511807077"},{"key":"atypb22","doi-asserted-by":"publisher","DOI":"10.1109\/TIT.1972.1054903"},{"key":"atypb23","doi-asserted-by":"publisher","DOI":"10.1109\/TIT.1969.1054304"},{"key":"atypb24","doi-asserted-by":"publisher","DOI":"10.1109\/TIT.2017.2672961"},{"key":"atypb25","doi-asserted-by":"publisher","DOI":"10.1109\/TIT.2018.2883615"},{"key":"atypb26","doi-asserted-by":"publisher","DOI":"10.1137\/16M1108431"},{"key":"atypb27","doi-asserted-by":"publisher","DOI":"10.1109\/TIT.2017.2743687"},{"key":"atypb28","first-page":"5699","volume":"63","author":"Li S.","year":"2017","journal-title":"IEEE Trans. Inform. Theory"},{"key":"atypb29","doi-asserted-by":"publisher","DOI":"10.1007\/s10623-019-00691-0"},{"key":"atypb30","doi-asserted-by":"publisher","DOI":"10.1016\/j.disc.2017.04.001"},{"key":"atypb31","doi-asserted-by":"publisher","DOI":"10.1016\/j.ffa.2018.09.005"},{"key":"atypb32","doi-asserted-by":"publisher","DOI":"10.1109\/TIT.2019.2932670"},{"key":"atypb33","unstructured":"Y. Liu, R. Li, L. Guo, and H. Song,\n                      Dimensions of Nonbinary Antiprimitive BCH Codes and Some Conjectures\n                      , preprint,https:\/\/arxiv.org\/abs\/1712.06842, 2018."},{"key":"atypb34","unstructured":"F. J. MacWilliams and N. J. A. Sloane,\n                      The Theory of Error-Correcting Codes\n                      , North-Holland, Amsterdam, 1977."},{"key":"atypb35","doi-asserted-by":"publisher","DOI":"10.1016\/S0019-9958(62)90298-X"},{"key":"atypb36","first-page":"9","author":"Peterson W. W.","year":"1969","journal-title":"NC"},{"key":"atypb37","doi-asserted-by":"crossref","unstructured":"V. Pless,\n                      Introduction to the Theory of Error-Correcting Codes\n                      , 3rd ed., John Wiley & Sons, New York, 1998.","DOI":"10.1002\/9781118032749"},{"key":"atypb38","first-page":"273","author":"Sloane N. J. A.","year":"2002","journal-title":"Berlin"},{"key":"atypb39","doi-asserted-by":"publisher","DOI":"10.1016\/j.ffa.2019.101581"},{"key":"atypb40","unstructured":"F. Yan,\n                      A Construction for $q$-ary Cyclic Codes with Large Minimum Distance\n                      , Master's thesis, Graduate School of Information Science, Nagoya University, 2012."},{"key":"atypb41","doi-asserted-by":"publisher","DOI":"10.1007\/s00200-017-0320-4"},{"key":"atypb42","doi-asserted-by":"publisher","DOI":"10.3934\/amc.2018034"},{"key":"atypb43","doi-asserted-by":"publisher","DOI":"10.1109\/18.887870"},{"key":"atypb44","doi-asserted-by":"publisher","DOI":"10.1007\/BF02685830"},{"key":"atypb45","doi-asserted-by":"publisher","DOI":"10.1109\/4234.784583"},{"key":"atypb46","doi-asserted-by":"publisher","DOI":"10.1109\/TIT.2019.2913389"}],"container-title":["SIAM Journal on Discrete Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/19M1260876","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:19:03Z","timestamp":1787321943000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/19M1260876"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,1]]},"references-count":46,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2021,1]]}},"alternative-id":["10.1137\/19M1260876"],"URL":"https:\/\/doi.org\/10.1137\/19m1260876","relation":{},"ISSN":["0895-4801","1095-7146"],"issn-type":[{"value":"0895-4801","type":"print"},{"value":"1095-7146","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,1]]}}}