This paper presented an implementation-oriented framework for constructing Deoxyribonucleic Acid (DNA) sequences using Bose-Chaudhuri-Hocquenghem (BCH) codes over the Eisenstein field $ \mathbb{Z}_2[\omega] $, where $ \omega $ was a primitive cube root of unity satisfying $ \omega^2+\omega+1 = 0 $. While $ \mathbb{Z}_2[\omega] $ was isomorphic to the familiar Galois field $ GF(4) $, we demonstrated that the Eisenstein representation offered distinct advantages for DNA coding applications. These included an intuitive algebraic structure that provided a mapping to Watson-Crick complementarity through the operation $ x \mapsto x+1 $, simple arithmetic operations that facilitated efficient hardware and software implementation, and a geometric interpretation via the complex plane that may enhance pedagogical accessibility. The four elements $ \{0, 1, \omega, 1+\omega\} $ were bijectively mapped to the nucleotide bases $ \{A, C, G, T\} $ in a manner that respected complementary pairing. We constructed BCH codes via field extensions $ \mathbb{Z}_2[\omega]^m $ based on irreducible polynomials, and presented a complete decoding algorithm adapted to the Eisenstein setting. The methodology incorporated biological constraints, including guanine–cytosine content (GC-content) balance and homopolymer restrictions. Experimental results demonstrated that our implementation achieved error correction performance identical to Galois field GF(4)-based BCH codes, as expected from the algebraic isomorphism, while offering implementation benefits in terms of arithmetic simplicity and reduced hardware resource utilization. Synthesis results on a Xilinx Artix-7 field-programmable gate array (FPGA) showed a 34% reduction in Look-Up Table (LUT) usage and a 17% improvement in maximum clock frequency for the Eisenstein-based decoder compared to the GF(4)-based implementation. Simulation results showed decoding success rates exceeding 99% for all tested configurations, with code rates ranging from 46.67% to 96.86%. The framework served as a bridge between number-theoretic constructions and practical DNA data storage systems.
Citation: Muhammad Sajjad, Nawaf A. Alqwaifly. DNA sequences from BCH codes over Eisenstein field and their decoding algorithm[J]. AIMS Mathematics, 2026, 11(10): 35529-35550. doi: 10.3934/math.20261378
This paper presented an implementation-oriented framework for constructing Deoxyribonucleic Acid (DNA) sequences using Bose-Chaudhuri-Hocquenghem (BCH) codes over the Eisenstein field $ \mathbb{Z}_2[\omega] $, where $ \omega $ was a primitive cube root of unity satisfying $ \omega^2+\omega+1 = 0 $. While $ \mathbb{Z}_2[\omega] $ was isomorphic to the familiar Galois field $ GF(4) $, we demonstrated that the Eisenstein representation offered distinct advantages for DNA coding applications. These included an intuitive algebraic structure that provided a mapping to Watson-Crick complementarity through the operation $ x \mapsto x+1 $, simple arithmetic operations that facilitated efficient hardware and software implementation, and a geometric interpretation via the complex plane that may enhance pedagogical accessibility. The four elements $ \{0, 1, \omega, 1+\omega\} $ were bijectively mapped to the nucleotide bases $ \{A, C, G, T\} $ in a manner that respected complementary pairing. We constructed BCH codes via field extensions $ \mathbb{Z}_2[\omega]^m $ based on irreducible polynomials, and presented a complete decoding algorithm adapted to the Eisenstein setting. The methodology incorporated biological constraints, including guanine–cytosine content (GC-content) balance and homopolymer restrictions. Experimental results demonstrated that our implementation achieved error correction performance identical to Galois field GF(4)-based BCH codes, as expected from the algebraic isomorphism, while offering implementation benefits in terms of arithmetic simplicity and reduced hardware resource utilization. Synthesis results on a Xilinx Artix-7 field-programmable gate array (FPGA) showed a 34% reduction in Look-Up Table (LUT) usage and a 17% improvement in maximum clock frequency for the Eisenstein-based decoder compared to the GF(4)-based implementation. Simulation results showed decoding success rates exceeding 99% for all tested configurations, with code rates ranging from 46.67% to 96.86%. The framework served as a bridge between number-theoretic constructions and practical DNA data storage systems.
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Figures(1) / Tables(8)
Muhammad Sajjad, Nawaf A. Alqwaifly. DNA sequences from BCH codes over Eisenstein field and their decoding algorithm[J]. AIMS Mathematics, 2026, 11(10): 35529-35550. doi: 10.3934/math.20261378
Geometric representation of $ \mathbb{Z}_2[\omega] $ in the complex plane. The four elements form a parallelogram, with addition corresponding to vector addition modulo 2 and multiplication by $ \omega $ corresponding to a $ 120^\circ $ rotation
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