Remarks on generalized toric codes

On the structure of generalized toric codes

Toric codes are obtained by evaluating rational functions of a nonsingular toric variety at the algebraic torus. One can extend toric codes to the so called generalized toric codes. This extension consists on evaluating elements of an arbitrary polynomial algebra at the algebraic torus instead of a linear combination of monomials whose exponents are rational points of a convex polytope. We study their multicyclic and metric structure, and we use them to express their dual and to estimate their minimum distance.

广义Toric码的子域子码

Subfield-Subcodes of Generalized Toric codes

本文研究了$\mathbb{F}_{p^s}$上广义Toric码的子域子码，通过确定其多项式生成元来计算维数并界定最小距离，并给出了达到已知最优性能的二进制和三进制示例。

广义毕达哥拉斯定理：通过环面几何研究某些对偶平坦空间的紧化

The generalized Pythagorean theorem on the compactifications of certain dually flat spaces via toric geometry

本文通过环面几何研究对偶平坦空间的紧化，在边界引入Bregman散度并证明其连续性与广义毕达哥拉斯定理，同时刻画了环面Kähler流形作为混合族环化的条件。

Generalized Hamming weights of toric codes over hypersimplices and square-free affine evaluation codes

Let $\mathbb{F}_{q}$ be a finite field with $q$ elements, where $q$ is a power of prime $p$. A polynomial over $\mathbb{F}_{q}$ is square-free if all its monomials are square-free. In this note, we determine an upper bound on the number of zeroes in the affine torus $T=(\mathbb{F}_{q}^{*})^{s}$ of any set of $r$ linearly independent square-free polynomials over $\mathbb{F}_{q}$ in $s$ variables, under certain conditions on $r$, $s$ and degree of these polynomials. Applying the results, we partly obtain the generalized Hamming weights of toric codes over hypersimplices and square-free evaluatio…

Quantum Codes from Toric Surfaces

A theory for constructing quantum error correcting codes from Toric surfaces by the Calderbank-Shor-Steane method is presented. In particular we study the method on toric Hirzebruch surfaces. The results are obtained by constructing a dualizing differential form for the toric surface and by using the cohomology and the intersection theory of toric varieties. In earlier work the author developed methods to construct linear error correcting codes from toric varieties and derive the code parameters using the cohomology and the intersection theory on toric varieties. This method is generalized in …

Lattice polytopes in coding theory

In this paper we discuss combinatorial questions about lattice polytopes motivated by recent results on minimum distance estimation for toric codes. We also prove a new inductive bound for the minimum distance of generalized toric codes. As an application, we give new formulas for the minimum distance of generalized toric codes for special lattice point configurations.