Characterizing the duals of linear codes with rich algebraic structures received great interest in recent decades. The beginning was by representing cyclic codes over finite fields as ideals in the polynomial ring. Subsequently, studying the duals of constacyclic, quasi-cyclic, quasi-twisted, generalized quasi-cyclic, and multi-twisted codes appeared extensively in literature. We consider the class of multi-twisted (MT) codes because it extends to all of these codes. We describe a MT code $\mathcal{C}$ as a module over a principal ideal domain. Hence, $\mathcal{C}$ has a generator polynomial matrix (GPM) that satisfies an identical equation. The reduced GPM of $\mathcal{C}$ is the Hermite normal form of its GPM. We show that the Euclidean dual $\mathcal{C}^\perp$ of $\mathcal{C}$ is MT as well. We prove a formula for a GPM of $\mathcal{C}^\perp$ using the identical equation of the reduced GPM of $\mathcal{C}$. Then we aim to replace the Euclidean dual with the Galois dual. The Galois inner product is an asymmetric form, so we distinguish between the right and left Galois duals. We show that the right and left Galois duals of a MT code are MT as well but with possibly different shift constants. Our study is the first to contain the right and left Galois duals of a linear code simultaneously. This gives two advantages: establishing their interconnected identities and introducing the two-sided Galois dual that has not previously appeared in the literature. We use a condition for the two-sided Galois dual of a MT code to be MT, hence its GPM is characterized. Two special cases are also studied, one when the right and left Galois duals trivially intersect and the other when they coincide. The latter case is considered for any linear code, where a necessary and sufficient condition is established for the equality of the right and left Galois duals.
翻译:具有丰富的代数结构的线性代码的双重特征在最近几十年里引起了极大的兴趣。 开始是代表以有限字段的周期性代码作为多边环中的理想。 因此, $\ mathal{ c} $ 具有一个能满足相同方程式的发电机多数值矩阵(GMM ) 。 降低的 $\ mathal{ c} 和多维代码是其GPM 的正常形式。 我们认为, 多维代码( MT) 的等级( MT) 因为它扩展到所有这些代码。 我们描述的是双双维代码( MT) $\ mathalcal{ calcal{C} 是一个模块。 我们证明, 双双基的双基代码是 美元( CMT), 其双基 Galthal_ cal_ cal_ cal_ cal_ cal_ cal_ deal_ dismal_ lax lax lax lax the legal_ gal_Gal_cal_cal_cal_cal dal_cal max max the modeal_cal_cal_ mocal_ modeal_cal_ modeal_ modeal_ modeal_ modeal_ modeal_ mocal_ mocal_ mocal_ mode modeal_ mocal_ mocal_ mocal_ modeal_ modeal_ modeal_ mocal_ mocal_ mocal_ mocal_ mocal_ mocal_ la la la mo mode modeal_ mocal mo modeal_ mo modeal mode modeal modeal mocal_ mocal mode mocal mocal mocal_ mod mo mo mode mo mo mo mo mo mocal mocal_ mocal mocal mocal mocal_ mocal mocal