{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,23]],"date-time":"2026-04-23T14:50:49Z","timestamp":1776955849220,"version":"3.51.4"},"reference-count":25,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2021,3,9]],"date-time":"2021-03-09T00:00:00Z","timestamp":1615248000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["61971321"],"award-info":[{"award-number":["61971321"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"the Fundamental Research Funds for the Central Universities"},{"name":"Guangxi Collaborative Innovation Center of Multi-source Information  Integration and Intelligent Processing"},{"name":"the Guangxi Bagui Scholar Teams for Innovation and Research Project"},{"name":"the Guangxi Talent Highland Project of Big Data Intelligence and Application"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In this paper, we study the entropy functions on extreme rays of the polymatroidal region which contain a matroid, i.e., matroidal entropy functions. We introduce variable strength orthogonal arrays indexed by a connected matroid M and positive integer v which can be regarded as expanding the classic combinatorial structure orthogonal arrays. It is interesting that they are equivalent to the partition-representations of the matroid M with degree v and the (M,v) almost affine codes. Thus, a synergy among four fields, i.e., information theory, matroid theory, combinatorial design, and coding theory is developed, which may lead to potential applications in information problems such as network coding and secret-sharing. Leveraging the construction of variable strength orthogonal arrays, we characterize all matroidal entropy functions of order n\u22645 with the exception of log10\u00b7U2,5 and logv\u00b7U3,5 for some v.<\/jats:p>","DOI":"10.3390\/e23030323","type":"journal-article","created":{"date-parts":[[2021,3,9]],"date-time":"2021-03-09T06:22:32Z","timestamp":1615270952000},"page":"323","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["Matroidal Entropy Functions: A Quartet of Theories of Information, Matroid, Design, and Coding"],"prefix":"10.3390","volume":"23","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5322-4783","authenticated-orcid":false,"given":"Qi","family":"Chen","sequence":"first","affiliation":[{"name":"State Key Laboratory of Integrated Service Networks, Xidian University, Xi\u2019 an 710071, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0360-0610","authenticated-orcid":false,"given":"Minquan","family":"Cheng","sequence":"additional","affiliation":[{"name":"Guangxi Key Lab of Multi-Source Information Mining &amp; Security, Guangxi Normal University, Guilin 541004, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Baoming","family":"Bai","sequence":"additional","affiliation":[{"name":"State Key Laboratory of Integrated Service Networks, Xidian University, Xi\u2019 an 710071, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,3,9]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1982","DOI":"10.1109\/18.641561","article-title":"A non-Shannon type conditional inequality of information quantities","volume":"43","author":"Zhang","year":"1997","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1016\/S0019-9958(78)91063-X","article-title":"Polymatroidal dependence structure of a set of random variables","volume":"39","author":"Fujishige","year":"1978","journal-title":"Inf. Contr."},{"key":"ref_3","unstructured":"Yeung, R.W. (2008). Information Theory and Network Coding, Springer."},{"key":"ref_4","first-page":"6","article-title":"Facets of entropy","volume":"62","author":"Yeung","year":"2012","journal-title":"IEEE Inf. Theory Soc."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"379","DOI":"10.3390\/e13020379","article-title":"Recent progresses in characterizing information inequalities","volume":"13","author":"Chan","year":"2011","journal-title":"Entropy"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1440","DOI":"10.1109\/18.681320","article-title":"On characterization of entropy function via information inequalities","volume":"44","author":"Zhang","year":"1998","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_7","first-page":"185","article-title":"Probabilistic conditional independence structures and matroid theory: Background","volume":"22","year":"1994","journal-title":"Int. J. Gen. Syst."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"169","DOI":"10.1016\/S0012-365X(99)00004-7","article-title":"Matroid representations by partitions","volume":"203","year":"1999","journal-title":"Discrete Math."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"179","DOI":"10.1023\/A:1008244215660","article-title":"Almost affine codes","volume":"14","author":"Simonis","year":"1998","journal-title":"Desings Codes Cryptogr."},{"key":"ref_10","unstructured":"Welsh, D.J.A. (1976). Matroid Theory, Academic Press."},{"key":"ref_11","unstructured":"Oxley, J.G. (1992). Matroid Theory, Oxford Univ. Press."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"355","DOI":"10.1090\/S0002-9947-1978-0491269-9","article-title":"Semimodular functions and combinatorial geometries","volume":"238","author":"Nguyen","year":"1978","journal-title":"Trans. AMS"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Hedayat, A.S., Sloane, N.J.A., and Stufken, J. (1999). Orthogonal Arrays: Theory and Applications, Springer.","DOI":"10.1007\/978-1-4612-1478-6"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Colbourn, C.J., and Dinitz, J.H. (2007). Handbook of Combinatorial Designs, CRC Press.","DOI":"10.1201\/9781420010541"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"236","DOI":"10.1016\/j.jcta.2009.06.002","article-title":"Constructions of new orthogonal arrays and covering arrays of strength three","volume":"117","author":"Ji","year":"2010","journal-title":"J. Comb. Theory Ser. A"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"270","DOI":"10.1016\/j.jcta.2010.03.013","article-title":"On the existence of orthogonal arrays OA(3,5,4n + 2)","volume":"118","author":"Yin","year":"2011","journal-title":"J. Comb. Theory Ser. A"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"155","DOI":"10.1090\/S0025-5718-1973-0419270-0","article-title":"A catalogue of combinatorial geometries","volume":"27","author":"Blackburn","year":"1973","journal-title":"Math. Comp."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1949","DOI":"10.1109\/TIT.2007.896862","article-title":"Networks, Matroids, and non-Shannon Information Inequalities","volume":"53","author":"Dougherty","year":"2007","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1007\/BF00196772","article-title":"On the classification of ideal secret sharing schemes","volume":"4","author":"Brickell","year":"1991","journal-title":"J. Cryptol."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1007\/s001459900036","article-title":"On matroid characterization of ideal secret sharing schemes","volume":"11","author":"Golic","year":"1998","journal-title":"J. Cryptol."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Blakley, G.R., and Kabatianski, G.A. (1995, January 27\u201331). General Perfect Secret Sharing Schemes. Proceedings of the CRYPTO \u201995: 15th Annual International Cryptology Conference, Santa Barbara, CA, USA.","DOI":"10.1007\/3-540-44750-4_29"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1023\/A:1024741108241","article-title":"A representation of a family of secret sharing matroids","volume":"30","author":"Ng","year":"2003","journal-title":"Des. Codes Cryptogr."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1007\/s10623-003-4192-1","article-title":"Secret sharing schemes with three or four minimal qualified subsets","volume":"34","year":"2005","journal-title":"Des. Codes Cryptogr."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Apte, J., Chen, Q., and Walsh, J.M. (2016, January 11\u201314). Symmetries in the Entropy Space. Proceedings of the IEEE Information Theory Workshop, Cambridge, UK.","DOI":"10.1109\/ITW.2016.7606794"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"5385","DOI":"10.1109\/TIT.2016.2600580","article-title":"Partition-Symmetrical Entropy Functions","volume":"62","author":"Chen","year":"2016","journal-title":"IEEE Trans. Inf. Theory"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/3\/323\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T05:35:23Z","timestamp":1760160923000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/3\/323"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,3,9]]},"references-count":25,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2021,3]]}},"alternative-id":["e23030323"],"URL":"https:\/\/doi.org\/10.3390\/e23030323","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,3,9]]}}}