Construction and Decoding of Error--Correcting Codes from Ideal Lattices of Finite Ternary Gamma Semirings
Chandrasekhar Gokavarapu, D. Madhusudhana Rao
TL;DR
This work addresses the limitations of binary and ring-based coding by introducing a new class of error-correcting codes constructed from the ideal lattices of finite commutative ternary Gamma-semirings (TGS). The core approach uses the ternary operation $[\cdot,\cdot,\cdot]$ and the absorption order $\oplus$ to define code constraints and a quotient-based syndrome decoding in the quotient TGS $T/I$, with code parameters governed by the index $|T/I|$ and the minimal nonzero elements of the ideal lattice. Key contributions include a canonical construction via $k$-ideals, a precise relation for code size and dimension $k = n\log_{|T|}|I| = n - n\log_{|T|}|T/I|$, a ternary-weight-based minimum distance, and a concrete finite-TGS example illustrating decoding. The results demonstrate a genuinely higher-arity, nonlinear coding framework with parameter sets unattainable by classical linear or semiring-based codes, offering new decoding mechanisms and potential applications in nonlinear and multi-parameter communication settings.
Abstract
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.