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Comput. Theory"],"published-print":{"date-parts":[[2020,12,31]]},"abstract":"<jats:p>\n            In a recent article, Kim and Kopparty (2017) gave a deterministic algorithm for the unique decoding problem for polynomials of bounded total degree over a general grid\n            <jats:italic>S<\/jats:italic>\n            <jats:sub>1<\/jats:sub>\n            \u00d7 \u2026 \u00d7\n            <jats:italic>S<\/jats:italic>\n            <jats:sub>m<\/jats:sub>\n            . We show that their algorithm can be adapted to solve the unique decoding problem for the general family of\n            <jats:italic>Downset codes<\/jats:italic>\n            . Here, a downset code is specified by a family\n            <jats:italic>D<\/jats:italic>\n            of monomials closed under taking factors: The corresponding code is the space of evaluations of all polynomials that can be written as linear combinations of monomials from\n            <jats:italic>D<\/jats:italic>\n            .\n          <\/jats:p>","DOI":"10.1145\/3417754","type":"journal-article","created":{"date-parts":[[2020,11,8]],"date-time":"2020-11-08T11:52:50Z","timestamp":1604836370000},"page":"1-11","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":1,"title":["Decoding Variants of Reed-Muller Codes over Finite Grids"],"prefix":"10.1145","volume":"12","author":[{"given":"Srikanth","family":"Srinivasan","sequence":"first","affiliation":[{"name":"Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai, Maharashtra, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Utkarsh","family":"Tripathi","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai, Maharashtra, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"S.","family":"Venkitesh","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai, Maharashtra, India"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"320","published-online":{"date-parts":[[2020,11,8]]},"reference":[{"key":"e_1_2_1_1_1","doi-asserted-by":"publisher","DOI":"10.1109\/TIT.2015.2462817"},{"key":"e_1_2_1_2_1","doi-asserted-by":"publisher","DOI":"10.1145\/278298.278306"},{"key":"e_1_2_1_3_1","doi-asserted-by":"publisher","DOI":"10.1007\/BF01200056"},{"key":"e_1_2_1_4_1","doi-asserted-by":"publisher","DOI":"10.1007\/BF01275486"},{"volume-title":"Welch","year":"1986","author":"Berlekamp Elwyn R.","key":"e_1_2_1_5_1"},{"volume-title":"David Cox, John Little, and Donal O\u2019Shea. 2007. Ideals, Varieties, and Algorithms","edition":"3","key":"e_1_2_1_6_1"},{"key":"e_1_2_1_7_1","doi-asserted-by":"publisher","DOI":"10.1016\/0020-0190(78)90067-4"},{"key":"e_1_2_1_8_1","doi-asserted-by":"publisher","DOI":"10.1007\/s10623-012-9680-8"},{"key":"e_1_2_1_9_1","doi-asserted-by":"publisher","DOI":"10.1109\/18.782097"},{"key":"e_1_2_1_10_1","unstructured":"Venkatesan Guruswami Atri Rudra and Madhu Sudan. 2019. Essential Coding Theory (draft). Available at https:\/\/cse.buffalo.edu\/faculty\/atri\/courses\/coding-theory\/book\/web-coding-book.pdf.  Venkatesan Guruswami Atri Rudra and Madhu Sudan. 2019. Essential Coding Theory (draft). 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