We introduce the first geometric construction of codes in the sum-rank metric, which we called linearized Algebraic Geometry codes, using quotients of the ring of Ore polynomials with coefficients in the function field of an algebraic curve. We study the parameters of these codes and give lower bounds for their dimension and minimum distance. Our codes exhibit quite good parameters, respecting a similar bound to Goppa's bound for Algebraic Geometry codes in the Hamming metric. Furthermore, our construction yields codes asymptotically better than the sum-rank version of the Gilbert-Varshamov bound.

翻译：我们首次提出了求和秩度量下码的几何构造，称为线性化代数几何码，该构造利用代数曲线函数域系数奥雷多项式环的商环实现。我们研究了这些码的参数，并给出了其维数和最小距离的下界。这些码展现出相当良好的参数特性，满足与汉明度量下代数几何码的戈帕界类似的界。此外，我们的构造得到的码在渐近性能上优于吉尔伯特-瓦尔沙莫夫界的求和秩版本。