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Codes Cryptogr."],"published-print":{"date-parts":[[2025,8]]},"abstract":"Abstract<\/jats:title>\n In this paper, we focus on constructing unique-decodable and list-decodable codes for the recently studied (t<\/jats:italic>,\u00a0e<\/jats:italic>)-composite-asymmetric error-correcting codes ((t<\/jats:italic>,\u00a0e<\/jats:italic>)-CAECCs). Let \n \n $$\\mathcal {X}$$<\/jats:tex-math>\n \n X<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> be an \n \n $$m \\times n$$<\/jats:tex-math>\n \n \n m<\/mml:mi>\n \u00d7<\/mml:mo>\n n<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> binary matrix in which each row has Hamming weight w<\/jats:italic>. If at most t<\/jats:italic> rows of \n \n $$\\mathcal {X}$$<\/jats:tex-math>\n \n X<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> contain errors, and in each erroneous row, there are at most e<\/jats:italic> occurrences of \n \n $$1 \\rightarrow 0$$<\/jats:tex-math>\n \n \n 1<\/mml:mn>\n \u2192<\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> errors, we say that a (t<\/jats:italic>,\u00a0e<\/jats:italic>)-composite-asymmetric error occurs in \n \n $$\\mathcal {X}$$<\/jats:tex-math>\n \n X<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>. For general values of m<\/jats:italic>,\u00a0n<\/jats:italic>,\u00a0w<\/jats:italic>,\u00a0t<\/jats:italic>, and e<\/jats:italic>, we propose new constructions of (t<\/jats:italic>,\u00a0e<\/jats:italic>)-CAECCs with redundancy at most \n \n $$(t-1)\\log (m) + O(1)$$<\/jats:tex-math>\n \n \n (<\/mml:mo>\n t<\/mml:mi>\n -<\/mml:mo>\n 1<\/mml:mn>\n )<\/mml:mo>\n log<\/mml:mo>\n (<\/mml:mo>\n m<\/mml:mi>\n )<\/mml:mo>\n +<\/mml:mo>\n O<\/mml:mi>\n (<\/mml:mo>\n 1<\/mml:mn>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, where O<\/jats:italic>(1) is independent of the code length m<\/jats:italic>. In particular, this yields a class of (2,\u00a0e<\/jats:italic>)-CAECCs that are optimal in terms of redundancy. When m<\/jats:italic> is a prime power, the redundancy can be further reduced to \n \n $$(t-1)\\log (m) - O(\\log (m))$$<\/jats:tex-math>\n \n \n (<\/mml:mo>\n t<\/mml:mi>\n -<\/mml:mo>\n 1<\/mml:mn>\n )<\/mml:mo>\n log<\/mml:mo>\n (<\/mml:mo>\n m<\/mml:mi>\n )<\/mml:mo>\n -<\/mml:mo>\n O<\/mml:mi>\n (<\/mml:mo>\n log<\/mml:mo>\n (<\/mml:mo>\n m<\/mml:mi>\n )<\/mml:mo>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>. To further increase the code size, we introduce a combinatorial object called a weak \n \n $$B_e$$<\/jats:tex-math>\n \n \n B<\/mml:mi>\n e<\/mml:mi>\n <\/mml:msub>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>-set. When \n \n $$e = w$$<\/jats:tex-math>\n \n \n e<\/mml:mi>\n =<\/mml:mo>\n w<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, we present an efficient encoding and decoding method for our codes. Finally, we explore potential improvements by relaxing the requirement of unique decoding to list-decoding. We show that when the list size is t<\/jats:italic>! or an exponential function of t<\/jats:italic>, there exist list-decodable (t<\/jats:italic>,\u00a0e<\/jats:italic>)-CAECCs with constant redundancy. When the list size is two, we construct list-decodable (3,\u00a02)-CAECCs with redundancy \n \n $$\\log (m) + O(1)$$<\/jats:tex-math>\n \n \n log<\/mml:mo>\n (<\/mml:mo>\n m<\/mml:mi>\n )<\/mml:mo>\n +<\/mml:mo>\n O<\/mml:mi>\n (<\/mml:mo>\n 1<\/mml:mn>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>.<\/jats:p>","DOI":"10.1007\/s10623-025-01634-8","type":"journal-article","created":{"date-parts":[[2025,5,15]],"date-time":"2025-05-15T11:14:33Z","timestamp":1747307673000},"page":"3437-3462","update-policy":"https:\/\/2.zoppoz.workers.dev:443\/https\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["More on codes for combinatorial composite DNA"],"prefix":"10.1007","volume":"93","author":[{"given":"Zuo","family":"Ye","sequence":"first","affiliation":[]},{"given":"Omer","family":"Sabary","sequence":"additional","affiliation":[]},{"given":"Ryan","family":"Gabrys","sequence":"additional","affiliation":[]},{"given":"Eitan","family":"Yaakobi","sequence":"additional","affiliation":[]},{"given":"Ohad","family":"Elishco","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,5,15]]},"reference":[{"key":"1634_CR1","doi-asserted-by":"publisher","first-page":"1229","DOI":"10.1038\/s41587-019-0240-x","volume":"37","author":"L Anavy","year":"2019","unstructured":"Anavy L., Vaknin I., Atar O., Amit R., Yakhini Z.: Data storage in DNA with fewer synthesis cycles using composite DNA letters. 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