PG（4，4）中的新量子帽
New quantum caps in PG(4,4)
摘要：
Calderbank、Rains、Shor和Sloane（见\cite{Sloane}）证明了在量子计算中纠错是可能的。量子稳定子码是二元射影空间中一类关于辛形式自正交的可加四元码。在\cite{Bierbra}中给出了一个几何描述，并引入了量子帽的概念.量子帽对应于距离为$d=4$的量子稳定子码在GF（4）上为线性时的特殊情况。本文回顾了从量子纠错到辛几何的转化，并研究了PG（4，4）中的量子码，其中我们构造了20，29，30，32，33，34，36和38个点的完全量子帽和10，12，13，20，23，24个点的不完全量子帽，25和26点，证明了11-量子帽的不存在性。具体地，大小为36和38的量子帽对在数据库\cite{codetable}中保持开放的量子码$[[36,26,4]]$和$[[38,28,4]]$的存在问题产生肯定的答案。
英文摘要：
Calderbank, Rains, Shor and Sloane (see \cite{Sloane}) showed that error-correction is possible in the context of quantum computations. Quantum stabilizer codes are a class of additive quaternary codes in binary projective spaces, which are self-orthogonal with respect to the symplectic form. A geometric description is given in \cite{Bierbra}, where also the notion of quantum cap is introduced. Quantum caps correspond to the special case of quantum stabilizer codes of distance $d=4$ when the code is linear over GF(4). In the present paper we review the translation from quantum error-correction to symplectic geometry and study quantum codes in PG(4,4) where we construct complete quantum caps with 20, 29, 30, 32, 33, 34, 36 and 38 points and incomplete quantum caps with 10, 12, 13, 20, 23, 24, 25 and 26 points and we prove the non existence of 11-quantum caps. In particular the quantum caps of sizes 36 and 38 yield positive answers to the existence questions of quantum codes $[[36,26,4]]$ and $[[38,28,4]]$ that remained open in the data base \cite{codetable}.
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