{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:56:14Z","timestamp":1760241374599,"version":"build-2065373602"},"reference-count":30,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2018,1,17]],"date-time":"2018-01-17T00:00:00Z","timestamp":1516147200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100002428","name":"Austrian Science Fund","doi-asserted-by":"publisher","award":["J 3765"],"award-info":[{"award-number":["J 3765"]}],"id":[{"id":"10.13039\/501100002428","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The generator matrices of polar codes and Reed\u2013Muller codes are submatrices of the Kronecker product of a lower-triangular binary square matrix. For polar codes, the submatrix is generated by selecting rows according to their Bhattacharyya parameter, which is related to the error probability of sequential decoding. For Reed\u2013Muller codes, the submatrix is generated by selecting rows according to their Hamming weight. In this work, we investigate the properties of the index sets selecting those rows, in the limit as the blocklength tends to infinity. We compute the Lebesgue measure and the Hausdorff dimension of these sets. We furthermore show that these sets are finely structured and self-similar in a well-defined sense, i.e., they have properties that are common to fractals.<\/jats:p>","DOI":"10.3390\/e20010070","type":"journal-article","created":{"date-parts":[[2018,1,17]],"date-time":"2018-01-17T12:17:11Z","timestamp":1516191431000},"page":"70","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["The Fractality of Polar and Reed\u2013Muller Codes"],"prefix":"10.3390","volume":"20","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3257-743X","authenticated-orcid":false,"given":"Bernhard","family":"Geiger","sequence":"first","affiliation":[{"name":"Signal Processing and Speech Communication Laboratory, Graz University of Technology, 8010 Graz, Austria"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2018,1,17]]},"reference":[{"key":"ref_1","unstructured":"Falconer, K. (2014). Fractal Geometry: Mathematical Foundations and Applications, John Wiley & Sons. [3rd ed.]."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Kahraman, S., Viterbo, E., and \u00c7elebi, M.E. (2013, January 2\u20134). Folded Tree Maximum-Likelihood Decoder for Kronecker Product-based Codes. Proceedings of the Allerton Conference on Communication, Control, and Computing, Monticello, IL, USA.","DOI":"10.1109\/Allerton.2013.6736584"},{"key":"ref_3","unstructured":"Abbe, E. Personal communication."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"447","DOI":"10.1109\/LCOMM.2008.080017","article-title":"A Performance Comparison of Polar Codes and Reed\u2013Muller Codes","volume":"12","year":"2008","journal-title":"IEEE Commun. Lett."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Renes, J.M., Sutter, D., and Hassani, S.H. (2015, January 14\u201319). Alignment of Polarized Sets. Proceedings of the 2015 IEEE International Symposium on Information Theory (ISIT), Hong Kong, China.","DOI":"10.1109\/ISIT.2015.7282895"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Bardet, M., Dragoi, V., Otmani, A., and Tillich, J.P. (arXiv, 2016). Algebraic Properties of Polar Codes From a New Polynomial Formalism, arXiv.","DOI":"10.1109\/ISIT.2016.7541295"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Bardet, M., Dragoi, V., Otmani, A., and Tillich, J.P. (2016, January 10\u201315). Algebraic properties of polar codes from a new polynomial formalism. Proceedings of the IEEE International Symposium on Information Theory (ISIT), Barcelona, Spain.","DOI":"10.1109\/ISIT.2016.7541295"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Haghighatshoar, S., and Abbe, E. (2013, January 7\u201312). Polarization of the R\u00e9nyi information dimension for single and multi terminal analog compression. Proceedings of the 2013 IEEE International Symposium on Information Theory Proceedings (ISIT), Istanbul, Turkey.","DOI":"10.1109\/ISIT.2013.6620332"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Abbe, E., and Wigderson, Y. (2015, January 14\u201319). High-Girth matrices and polarization. Proceedings of the IEEE International Symposium on Information Theory Proceedings (ISIT), Hong Kong, China.","DOI":"10.1109\/ISIT.2015.7282898"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"6931","DOI":"10.1109\/TIT.2016.2616642","article-title":"An Ergodic Theory of Binary Operations\u2014Part I: Key Properties","volume":"62","author":"Nasser","year":"2016","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1063","DOI":"10.1109\/TIT.2016.2617958","article-title":"An Ergodic Theory of Binary Operations\u2014Part II: Applications to Polarization","volume":"63","author":"Nasser","year":"2017","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Taylor, M. (2006). Measure Theory and Integration, American Mathematical Society.","DOI":"10.1090\/gsm\/076"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"3051","DOI":"10.1109\/TIT.2009.2021379","article-title":"Channel polarization: A method for constructing capacity-achieving codes for symmetric binary-input memoryless channels","volume":"55","year":"2009","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1751","DOI":"10.1109\/TIT.2010.2040961","article-title":"Polar Codes are Optimal for Lossy Source Coding","volume":"56","author":"Korada","year":"2010","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"259","DOI":"10.1561\/0100000041","article-title":"Polarization and Polar Codes","volume":"8","year":"2011","journal-title":"Found. Trends Commun. Inf. Theory"},{"key":"ref_16","unstructured":"Korada, S.B. (2009). Polar Codes for Channel and Source Coding. [Ph.D. Thesis, \u00c9cole Polytechnique F\u00e9d\u00e9rale de Lausanne]."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"6562","DOI":"10.1109\/TIT.2013.2272694","article-title":"How to Construct Polar Codes","volume":"59","author":"Tal","year":"2013","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"El-Khamy, M., Mahdavifar, H., Feygin, G., Lee, J., and Kang, I. (2015, January 9\u201312). Relaxed channel polarization for reduced complexity polar coding. Proceedings of the IEEE Wireless Communications and Networking Conference (WCNC), New Orleans, LA, USA.","DOI":"10.1109\/WCNC.2015.7127470"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1986","DOI":"10.1109\/TIT.2017.2657691","article-title":"Relaxed Polar Codes","volume":"63","author":"Mahdavifar","year":"2017","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_20","unstructured":"Hassani, S.H., Korada, S., and Urbanke, R. (October, January 30). The compound capacity of polar codes. Proceedings of the Allerton Conference on Communication, Control, and Computing, Monticello, IL, USA."},{"key":"ref_21","unstructured":"Kuipers, I., and Niederreiter, H. (1974). Uniform Distribution of Sequences, John Wiley & Sons."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"615","DOI":"10.1016\/j.bulsci.2004.12.004","article-title":"Topological and fractal properties of real numbers which are not normal","volume":"129","author":"Albeverio","year":"2005","journal-title":"Bull. Sci. Math."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1109\/TIT.1968.1054127","article-title":"New generalizations of the Reed\u2013Muller codes\u2013I: Primitive codes","volume":"14","author":"Kasami","year":"1968","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"403","DOI":"10.1016\/S0019-9958(70)90214-7","article-title":"On generalized Reed\u2013Muller codes and their relatives","volume":"16","author":"Delsarte","year":"1970","journal-title":"Inf. Control"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"5875","DOI":"10.1109\/TIT.2014.2341919","article-title":"Finite-Length Scaling for Polar Codes","volume":"60","author":"Hassani","year":"2014","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Parthasarathy, K.R. (1967). Probability Measures on Metric Spaces, Academic Press.","DOI":"10.1016\/B978-1-4832-0022-4.50007-7"},{"key":"ref_27","unstructured":"Rudin, W. (1987). Real and Complex Analysis, McGraw-Hill. [3rd ed.]."},{"key":"ref_28","unstructured":"Feller, W. (1968). An Introduction to Probability Theory and Its Applications, John Wiley & Sons. [3rd ed.]."},{"key":"ref_29","unstructured":"Abramowitz, M., and Stegun, I.A. (1972). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover Publications. [9th ed.]."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1093\/qmath\/os-20.1.31","article-title":"The Fractional Dimension of a Set defined by decimal properties","volume":"os-20","author":"Eggleston","year":"1949","journal-title":"Q. J. Math."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/20\/1\/70\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T14:51:40Z","timestamp":1760194300000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/20\/1\/70"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,1,17]]},"references-count":30,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2018,1]]}},"alternative-id":["e20010070"],"URL":"https:\/\/doi.org\/10.3390\/e20010070","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2018,1,17]]}}}