We introduce a new wide class of error-correcting codes, called non-split toric codes. These codes are a natural generalization of toric codes where non-split algebraic tori are taken instead of usual (i.e., split) ones. The main advantage of the new codes is their cyclicity; hence, they can possibly be decoded quite fast. Many classical codes, such as (doubly-extended) Reed-Solomon and (projective) Reed-Muller codes, are contained (up to equivalence) in the new class. Our codes are explicitly described in terms of algebraic and toric geometries over finite fields; therefore, they can easily be constructed in practice. Finally, we obtain new cyclic reversible codes, namely non-split toric codes on the del Pezzo surface of degree 6 and Picard number 1. We also compute their parameters, which prove to attain current lower bounds at least for small finite fields.

中文翻译：

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非拆分复曲面代码

我们介绍了一种新型的纠错码，称为非拆分复曲面码。这些代码是复曲面代码的自然概括，其中采用非拆分代数花托，而不是通常的（即拆分）代数。新代码的主要优点是它们的周期性。因此，它们可能会很快解码。新类中包含（直到等效）许多经典代码，例如（双扩展）Reed-Solomon和（投射）Reed-Muller码。我们的代码是根据有限域上的代数和复曲面几何来明确描述的；因此，它们可以在实践中轻松构建。最后，我们获得了新的循环可逆代码，即6度和皮卡德数为1的del Pezzo表面上的非分裂复曲面代码。我们还计算了它们的参数，