In this paper, we study algebraic geometry codes from curves over $\mathbb{F}_{q^\ell}$ through their virtual projections which are algebraic geometric codes over $\mathbb{F}_q$. We use the virtual projections to provide fractional decoding algorithms for the codes over $\mathbb{F}_{q^\ell}$. Fractional decoding seeks to perform error correction using a smaller fraction of $\mathbb{F}_q$-symbols than a typical decoding algorithm. In one instance, the bound on the number of correctable errors differs from the usual lower bound by the degree of a pole divisor of an annihilator function. In another, we view the virtual projections as interleaved codes to, with high probability, correct more errors than anticipated.

翻译：本文研究来自曲线在 $\mathbb{F}_{q^\ell}$ 上的代数几何码，通过其虚拟投影（即 $\mathbb{F}_q$ 上的代数几何码）进行分析。我们利用虚拟投影为 $\mathbb{F}_{q^\ell}$ 上的码提出分式译码算法。分式译码旨在使用比传统译码算法更少的 $\mathbb{F}_q$ 符号分数来执行纠错。在一种情形中，可纠错错误数量的界限与通常下界的差异等于消零函数极点除子的度数。在另一情形中，我们将虚拟投影视为交织码，以高概率纠正超出预期的更多错误。