{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,7]],"date-time":"2026-02-07T21:03:20Z","timestamp":1770498200506,"version":"3.49.0"},"reference-count":36,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2024,7,9]],"date-time":"2024-07-09T00:00:00Z","timestamp":1720483200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2024,7,9]],"date-time":"2024-07-09T00:00:00Z","timestamp":1720483200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["AAECC"],"published-print":{"date-parts":[[2026,1]]},"abstract":"Abstract<\/jats:title>\n \n Let\n \n \n $${\\mathfrak {R}}= {\\mathbb {Z}}_4[u,v]\/\\langle u^2-2,uv-2,v^2,2u,2v\\rangle$$<\/jats:tex-math>\n \n \n R<\/mml:mi>\n =<\/mml:mo>\n \n Z<\/mml:mi>\n 4<\/mml:mn>\n <\/mml:msub>\n \n [<\/mml:mo>\n u<\/mml:mi>\n ,<\/mml:mo>\n v<\/mml:mi>\n ]<\/mml:mo>\n <\/mml:mrow>\n \/<\/mml:mo>\n \n \u27e8<\/mml:mo>\n \n u<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n -<\/mml:mo>\n 2<\/mml:mn>\n ,<\/mml:mo>\n u<\/mml:mi>\n v<\/mml:mi>\n -<\/mml:mo>\n 2<\/mml:mn>\n ,<\/mml:mo>\n \n v<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n ,<\/mml:mo>\n 2<\/mml:mn>\n u<\/mml:mi>\n ,<\/mml:mo>\n 2<\/mml:mn>\n v<\/mml:mi>\n \u27e9<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n be a ring, where\n \n \n $${\\mathbb {Z}}_{4}$$<\/jats:tex-math>\n \n \n Z<\/mml:mi>\n 4<\/mml:mn>\n <\/mml:msub>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n is a ring of integers modulo 4. This ring\n \n \n $${\\mathfrak {R}}$$<\/jats:tex-math>\n \n R<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n is a local non-chain ring of characteristic 4. The main objective of this article is to construct reversible cyclic codes of odd length\n n<\/jats:italic>\n over the ring\n \n \n $${\\mathfrak {R}}.$$<\/jats:tex-math>\n \n \n R<\/mml:mi>\n .<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n Employing these reversible cyclic codes, we obtain reversible cyclic DNA codes of length\n n<\/jats:italic>\n ,\u00a0 based on the deletion distance over the ring\n \n \n $${\\mathfrak {R}}.$$<\/jats:tex-math>\n \n \n R<\/mml:mi>\n .<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n We also construct a bijection\n \n \n $$\\Gamma$$<\/jats:tex-math>\n \n \u0393<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n between the elements of the ring\n \n \n $${\\mathfrak {R}}$$<\/jats:tex-math>\n \n R<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n and\n \n \n $$S_{D_{16}}.$$<\/jats:tex-math>\n \n \n \n S<\/mml:mi>\n \n D<\/mml:mi>\n 16<\/mml:mn>\n <\/mml:msub>\n <\/mml:msub>\n .<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n As an application of\n \n \n $$\\Gamma ,$$<\/jats:tex-math>\n \n \n \u0393<\/mml:mi>\n ,<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n the reversibility problem which occurs in DNA\n k<\/jats:italic>\n -bases has been solved. Moreover, we introduce a Gray map\n \n \n $$\\Psi _{\\hom }:{\\mathfrak {R}}^{n}\\rightarrow {\\mathbb {F}}_{2}^{8n}$$<\/jats:tex-math>\n \n \n \n \u03a8<\/mml:mi>\n hom<\/mml:mo>\n <\/mml:msub>\n :<\/mml:mo>\n \n \n R<\/mml:mi>\n <\/mml:mrow>\n n<\/mml:mi>\n <\/mml:msup>\n \u2192<\/mml:mo>\n \n F<\/mml:mi>\n \n 2<\/mml:mn>\n <\/mml:mrow>\n \n 8<\/mml:mn>\n n<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msubsup>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n with respect to homogeneous weight\n \n \n $$w_{\\hom }$$<\/jats:tex-math>\n \n \n w<\/mml:mi>\n hom<\/mml:mo>\n <\/mml:msub>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n over the ring\n \n \n $${\\mathfrak {R}}$$<\/jats:tex-math>\n \n R<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n . Further, we discuss the\n GC<\/jats:italic>\n -content of DNA cyclic codes and their deletion distance. 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