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Non-split Toric Codes
Problems of Information Transmission ( IF 0.5 ) Pub Date : 2019-07-12 , DOI: 10.1134/s0032946019020029
D. I. Koshelev

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:

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,
更新日期:2019-07-12
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