{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:56:29Z","timestamp":1760147789297,"version":"build-2065373602"},"reference-count":33,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2023,3,1]],"date-time":"2023-03-01T00:00:00Z","timestamp":1677628800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Science Foundation of China","award":["62141101"],"award-info":[{"award-number":["62141101"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>As a typical symbol-wise solution of asymmetric Slepian-Wolf coding problem, Distributed Arithmetic Coding (DAC) non-linearly partitions source space into disjoint cosets with unequal sizes. The distribution of DAC coset cardinalities, named the Coset Cardinality Spectrum (CCS), plays an important role in both theoretical understanding and decoder design for DAC. In general, CCS cannot be calculated directly. Instead, a numerical algorithm is usually used to obtain an approximation. This paper first finds that the contemporary numerical algorithm of CCS is theoretically imperfect and does not finally converge to the real CCS. Further, to solve this problem, we refine the original numerical algorithm based on rigorous theoretical analyses. Experimental results verify that the refined numerical algorithm amends the drawbacks of the original version.<\/jats:p>","DOI":"10.3390\/e25030437","type":"journal-article","created":{"date-parts":[[2023,3,2]],"date-time":"2023-03-02T03:06:43Z","timestamp":1677726403000},"page":"437","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Fair Numerical Algorithm of Coset Cardinality Spectrum for Distributed Arithmetic Coding"],"prefix":"10.3390","volume":"25","author":[{"given":"Yong","family":"Fang","sequence":"first","affiliation":[{"name":"School of Information Engineering, Chang\u2019an University, Xi\u2019an 710064, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2012-6861","authenticated-orcid":false,"given":"Nan","family":"Yang","sequence":"additional","affiliation":[{"name":"School of Information Engineering, Chang\u2019an University, Xi\u2019an 710064, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,3,1]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"471","DOI":"10.1109\/TIT.1973.1055037","article-title":"Noiseless coding of correlated information sources","volume":"19","author":"Slepian","year":"1973","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_2","unstructured":"Longo, G. (1977). The Information Theory Approach to Communications, Springer."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1109\/TIT.1976.1055508","article-title":"The rate-distortion function for source coding with side information at the decoder","volume":"22","author":"Wyner","year":"1976","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"60","DOI":"10.1016\/S0019-9958(78)90034-7","article-title":"The rate-distortion function for source coding with side information at the decoder-II: General sources","volume":"38","author":"Wyner","year":"1978","journal-title":"Inf. Control"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1181","DOI":"10.1109\/TIT.2003.810622","article-title":"Duality between source coding and channel coding and its extension to the side information case","volume":"49","author":"Pradhan","year":"2003","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"4006","DOI":"10.1109\/TIT.2009.2025527","article-title":"On the duality between Slepian-Wolf coding and channel coding under mismatched decoding","volume":"55","author":"Chen","year":"2009","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"5575","DOI":"10.1109\/TIT.2009.2032815","article-title":"On the linear codebook-level duality between Slepian-Wolf coding and channel coding","volume":"55","author":"Chen","year":"2009","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"2534","DOI":"10.1109\/TCOMM.2009.09.080018","article-title":"The equivalence between Slepian-Wolf coding and channel coding under density evolution","volume":"57","author":"Chen","year":"2009","journal-title":"IEEE Trans. Commun."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Chen, J., He, D.-K., Jagmohan, A., and Lastras-Montano, L.A. (2017). On the reliability function of variable-rate Slepian-Wolf coding. Entropy, 19.","DOI":"10.3390\/e19080389"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"417","DOI":"10.1109\/4234.957380","article-title":"Compression of correlated binary sources using turbo codes","volume":"5","author":"Zhao","year":"2001","journal-title":"IEEE Commun. Lett."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"440","DOI":"10.1109\/LCOMM.2002.804244","article-title":"Compression of binary sources with side information at the decoder using LDPC codes","volume":"6","author":"Liveris","year":"2002","journal-title":"IEEE Commun. Lett."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Bilkent, E. (2012, January 28\u201331). Polar coding for the Slepian-Wolf problem based on monotone chain rules. Proceedings of the IEEE International Symposium on Information Theory and Its Applications (ISITA2012), Honolulu, HI, USA.","DOI":"10.1109\/ISIT.2012.6284254"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"198","DOI":"10.1147\/rd.203.0198","article-title":"Generalized Kraft inequality and arithmetic coding","volume":"20","author":"Rissanen","year":"1976","journal-title":"IBM J. Res. Dev."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"520","DOI":"10.1145\/214762.214771","article-title":"Arithmetic coding for data compression","volume":"30","author":"Witten","year":"1987","journal-title":"Commun. ACM"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1109\/26.554275","article-title":"Integrating error detection into arithmetic coding","volume":"45","author":"Boyd","year":"1997","journal-title":"IEEE Trans. Commun."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1540","DOI":"10.1109\/26.950341","article-title":"Continuous error detection (CED) for reliable communication","volume":"49","author":"Anand","year":"2001","journal-title":"IEEE Trans. Commun."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1007","DOI":"10.1109\/TCOMM.2005.849690","article-title":"Joint source\/channel coding and MAP decoding of arithmetic codes","volume":"53","author":"Grangetto","year":"2005","journal-title":"IEEE Trans. Commun."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"4154","DOI":"10.1109\/TSP.2009.2023359","article-title":"Distributed coding using punctured quasi-arithmetic codes for memory and memoryless sources","volume":"57","author":"Malinowski","year":"2009","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"883","DOI":"10.1109\/LCOMM.2007.071172","article-title":"Distributed arithmetic coding","volume":"11","author":"Grangetto","year":"2007","journal-title":"IEEE Commun. Lett."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"2245","DOI":"10.1109\/TSP.2009.2014280","article-title":"Distributed arithmetic coding for the Slepian-Wolf problem","volume":"57","author":"Grangetto","year":"2009","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_21","first-page":"9","article-title":"Overlapped quasi-arithmetic codes for distributed video coding","volume":"II","author":"Artigas","year":"2007","journal-title":"Proc. IEEE ICIP"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Yang, N., Fang, Y., Wang, L., Wang, Z., and Jiang, F. (2022). Approximation of initial coset cardinality spectrum of distributed arithmetic coding for uniform binary sources. IEEE Commun. Lett., in progress to appear.","DOI":"10.1109\/LCOMM.2022.3219122"},{"key":"ref_23","unstructured":"Fang, Y., and Jeong, J. (2008). Distributed arithmetic coding for sources with hidden Markov correlation. arXiv."},{"key":"ref_24","unstructured":"Fang, Y. Q-ary distributed arithmetic coding for uniform Q-ary sources. IEEE Trans. Inf. Theory, in progress to appear."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"2750","DOI":"10.1109\/JSEN.2013.2257944","article-title":"Distributed block arithmetic coding for equiprobable sources","volume":"13","author":"Zhou","year":"2013","journal-title":"IEEE Sens. J."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Wang, Z., Mao, Y., and Kiringa, I. (2015, January 6\u20139). Non-binary distributed arithmetic coding. Proceedings of the IEEE 14th Canadian Workshop Information Theory (CWIT), St. John\u2019s, NL, Canada.","DOI":"10.1109\/CWIT.2015.7255140"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1079","DOI":"10.1109\/LSP.2009.2030099","article-title":"Distribution of distributed arithmetic codewords for equiprobable binary sources","volume":"16","author":"Fang","year":"2009","journal-title":"IEEE Signal Process. Lett."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1584","DOI":"10.1109\/TCOMM.2013.012913.120486","article-title":"DAC spectrum of binary sources with equally-likely symbols","volume":"61","author":"Fang","year":"2013","journal-title":"IEEE Trans. Commun."},{"key":"ref_29","first-page":"4305","article-title":"Analysis on tailed distributed arithmetic codes for uniform binary sources","volume":"64","author":"Fang","year":"2016","journal-title":"IEEE Trans. Commun."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"6580","DOI":"10.1109\/TIT.2020.3014965","article-title":"Codebook cardinality spectrum of distributed arithmetic coding for independent and identically-distributed binary sources","volume":"66","author":"Fang","year":"2020","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"256","DOI":"10.1109\/TCOMM.2013.112613.120796","article-title":"Improved binary DAC codec with spectrum for equiprobable sources","volume":"62","author":"Fang","year":"2014","journal-title":"IEEE Trans. Commun."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"8335","DOI":"10.1109\/TIT.2021.3116082","article-title":"Two applications of coset cardinality spectrum of distributed arithmetic coding","volume":"67","author":"Fang","year":"2021","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_33","unstructured":"(2022, December 15). GitHub. Available online: https:\/\/github.com\/fy79\/dac_ccs_num."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/3\/437\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T18:45:16Z","timestamp":1760121916000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/3\/437"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,3,1]]},"references-count":33,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2023,3]]}},"alternative-id":["e25030437"],"URL":"https:\/\/doi.org\/10.3390\/e25030437","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2023,3,1]]}}}