The main objective of this paper is to introduce a new approach for constructing reversible and reversible DNA codes over the finite fields $ F_{4^{2m}} $, $ m\ge 1 $ from any given polynomials by utilizing $ T^n $-set codes. Notably, these polynomials are not required to be self-reciprocal divisors of $ x^n-1 $. In addition to this method, we provide some results that demonstrate how to generate reversible and reversible DNA codes from any $ [n, k, d] $-cyclic codes over $ F_{4^{2m}} $ by applying $ T^n $-set codes. Moreover, this approach allows us to determine a lower bound for the distance before completing the entire calculation process. We also construct a correspondence table between DNA sequences made up of $ 20 $-bases and all $ 256 $ elements of the linear code $ \langle T^5_{g} \mid T^5_{(g^{*})^{\circ 4}}\rangle $ over $ F_{16} $, where $ g = w^{13} + x + w^2 x^2 +w^3 x^3 $ is a polynomial in $ x $ over $ F_{16} $ and $ (g^{*})^{\circ 4} $ represents the Hadamard $ 4^{th} $-power of the reciprocal polynomial $ g^{*} $ to demonstrate the fact that non-reversible codes may correspond to reversible DNA codes. Additionally, we provide a detailed table presenting some outcomes related to $ l $-MDS codes, self-orthogonal codes, and their associated parameters.
Citation: Elif Segah Oztas, Amal S. Alali, Shakir Ali, Mohd Azeem, Muhammad S. M. Asri, Kok Bin Wong. On the construction of reversible DNA codes over $ F_{4^{2m}} $ via $ T^n $-set codes[J]. AIMS Mathematics, 2025, 10(12): 29424-29453. doi: 10.3934/math.20251292
The main objective of this paper is to introduce a new approach for constructing reversible and reversible DNA codes over the finite fields $ F_{4^{2m}} $, $ m\ge 1 $ from any given polynomials by utilizing $ T^n $-set codes. Notably, these polynomials are not required to be self-reciprocal divisors of $ x^n-1 $. In addition to this method, we provide some results that demonstrate how to generate reversible and reversible DNA codes from any $ [n, k, d] $-cyclic codes over $ F_{4^{2m}} $ by applying $ T^n $-set codes. Moreover, this approach allows us to determine a lower bound for the distance before completing the entire calculation process. We also construct a correspondence table between DNA sequences made up of $ 20 $-bases and all $ 256 $ elements of the linear code $ \langle T^5_{g} \mid T^5_{(g^{*})^{\circ 4}}\rangle $ over $ F_{16} $, where $ g = w^{13} + x + w^2 x^2 +w^3 x^3 $ is a polynomial in $ x $ over $ F_{16} $ and $ (g^{*})^{\circ 4} $ represents the Hadamard $ 4^{th} $-power of the reciprocal polynomial $ g^{*} $ to demonstrate the fact that non-reversible codes may correspond to reversible DNA codes. Additionally, we provide a detailed table presenting some outcomes related to $ l $-MDS codes, self-orthogonal codes, and their associated parameters.
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Elif Segah Oztas, Amal S. Alali, Shakir Ali, Mohd Azeem, Muhammad S. M. Asri, Kok Bin Wong. On the construction of reversible DNA codes over $ F_{4^{2m}} $ via $ T^n $-set codes[J]. AIMS Mathematics, 2025, 10(12): 29424-29453. doi: 10.3934/math.20251292
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