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.

翻译：本文引入了一类新的纠错码，其构造基于有限交换三元伽马半环（TGS）的理想格。与依赖二元运算的经典线性码或环线性码不同，TGS码源于内在的三元运算 $[x,y,z]$ 以及控制坐标吸收的op-plus序。TGS码的基本参数由底层半环的$k$-理想结构决定：维数由指数 $|T/I|$ 给出，而最小距离取决于分配理想格 $L(T)$ 的最小非零元素。这导致参数集在有限域、群代数或标准半环框架下无法实现。本文发展了一种基于商集的译码方法，其中三元校验子 $S(c) = \\Phi(c) + I$ 位于商TGS $T/I$ 中，并将环境空间划分为由理想吸收决定的陪集。最小非零格元素产生规范错误代表元，从而形成一种类似于经典校验子译码但源于更高元交互的译码流程。通过一个具体的有限示例，阐明了参数计算、校验子类结构以及译码方法的性能。这些结果表明，三元伽马半环为非线性、非二进制及更高元编码理论提供了新的代数基础。其理想格结构与三元商行为催生了新的译码机制与错误模式，将代数编码理论扩展至经典线性系统的局限之外。