Construction and Decoding of Error--Correcting Codes from Ideal Lattices of Finite Ternary Gamma Semirings

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.