On the parameters of r-dimensional toric codes
On the parameters of r-dimensional toric codes
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DOI：
10.1016/j.ffa.2007.02.002
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发表时间：
2005-12
期刊：
影响因子：
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通讯作者：
D. Ruano
中科院分区：
文献类型：
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作者：
D. Ruano
J.P.Hansen从r⩾2的有理凸多面体出发，在有限域Fq上构造了一个长度为n=(q−1)的纠错码.有理凸多面体与正规环面簇和卡地亚因子是相同的基准。该码是在代数环面上对多面体定义的环面簇的有理函数进行赋值的，是Goppa意义下的赋值码。我们利用上同调来计算码的维度。利用交集理论和混合体理论估计了最小距离，推广了J.P.Hansen关于平面多面体的方法。最后，我们给出了对Joyner猜想的反例[D.Joyner，有限域上的Toric码，应用.代数工程。通讯。电脑。15(2004)63-79]。
From a rational convex polytope of dimension r⩾2 J.P. Hansen constructed an error correcting code of length n=(q−1)rover the finite field Fq. A rational convex polytope is the same datum as a normal toric variety and a Cartier divisor. The code is obtained evaluating rational functions of the toric variety defined by the polytope at the algebraic torus, and it is an evaluation code in the sense of Goppa. We compute the dimension of the code using cohomology. The minimum distance is estimated using intersection theory and mixed volumes, extending the methods of J.P. Hansen for plane polytopes. Finally we give counterexamples to Joyner's conjectures [D. Joyner, Toric codes over finite fields, Appl. Algebra Engrg. Comm. Comput. 15 (2004) 63–79].