Construction and Decoding of Error--Correcting Codes from Ideal Lattices of Finite Ternary Gamma Semirings
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| Format: | Preprint |
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2025
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| _version_ | 1866914169416056832 |
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| author | Gokavarapu, Chandrasekhar Rao, D. Madhusudhana |
| author_facet | Gokavarapu, Chandrasekhar Rao, D. Madhusudhana |
| contents | This paper introduces a new class of error-correcting codes constructed from the ideal lattices of finite commutative ternary Gamma-semirings (TGS). Unlike classical linear or ring-linear codes, which rely on binary operations, TGS codes arise from the intrinsic ternary operation $[x,y,z]$ and the op-plus order that governs coordinatewise absorption. The fundamental parameters of a TGS code are determined by the $k$-ideal structure of the underlying semiring: the dimension is given by the index $|T/I|$, while the minimum distance depends on the minimal nonzero elements of the distributive ideal lattice $L(T)$. This leads to parameter sets that are not achievable over finite fields, group algebras, or standard semiring frameworks.
A quotient-based decoding method is developed in which the ternary syndrome $S(c) = Phi(c) + I$ lies in the quotient TGS $T/I$ and partitions the ambient space into cosets determined by ideal absorption. Minimal nonzero lattice elements yield canonical error representatives, producing a decoding procedure that resembles classical syndrome decoding but comes from higher-arity interactions. A concrete finite example illustrates the computation of parameters, the structure of syndrome classes, and the performance of the decoding method.
These results show that ternary Gamma-semirings provide a new algebraic foundation for nonlinear, nonbinary, and higher-arity coding theory. Their ideal-lattice structure and ternary quotient behavior generate new decoding mechanisms and error profiles, expanding algebraic coding theory beyond the limitations of classical linear systems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_19378 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Construction and Decoding of Error--Correcting Codes from Ideal Lattices of Finite Ternary Gamma Semirings Gokavarapu, Chandrasekhar Rao, D. Madhusudhana Rings and Algebras Information Theory 16Y60, 20N15, 94B05, 94B60, 94A60, 08A30, 06B23 This paper introduces a new class of error-correcting codes constructed from the ideal lattices of finite commutative ternary Gamma-semirings (TGS). Unlike classical linear or ring-linear codes, which rely on binary operations, TGS codes arise from the intrinsic ternary operation $[x,y,z]$ and the op-plus order that governs coordinatewise absorption. The fundamental parameters of a TGS code are determined by the $k$-ideal structure of the underlying semiring: the dimension is given by the index $|T/I|$, while the minimum distance depends on the minimal nonzero elements of the distributive ideal lattice $L(T)$. This leads to parameter sets that are not achievable over finite fields, group algebras, or standard semiring frameworks. A quotient-based decoding method is developed in which the ternary syndrome $S(c) = Phi(c) + I$ lies in the quotient TGS $T/I$ and partitions the ambient space into cosets determined by ideal absorption. Minimal nonzero lattice elements yield canonical error representatives, producing a decoding procedure that resembles classical syndrome decoding but comes from higher-arity interactions. A concrete finite example illustrates the computation of parameters, the structure of syndrome classes, and the performance of the decoding method. These results show that ternary Gamma-semirings provide a new algebraic foundation for nonlinear, nonbinary, and higher-arity coding theory. Their ideal-lattice structure and ternary quotient behavior generate new decoding mechanisms and error profiles, expanding algebraic coding theory beyond the limitations of classical linear systems. |
| title | Construction and Decoding of Error--Correcting Codes from Ideal Lattices of Finite Ternary Gamma Semirings |
| topic | Rings and Algebras Information Theory 16Y60, 20N15, 94B05, 94B60, 94A60, 08A30, 06B23 |
| url | https://arxiv.org/abs/2511.19378 |