Small-bias sets from extended norm-trace codes
Small-bias sets from extended norm-trace codes
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来自扩展范数跟踪代码的小偏差集
DOI：
10.1090/conm/579/11526
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发表时间：
2011
影响因子：
1
通讯作者：
Justin D. Peachey
中科院分区：
文献类型：
--
作者：
Gretchen L. Matthews;Justin D. Peachey
正如Naor和Naor [9]以及其他人[1，2]所证明的那样，小偏差概率空间或小偏差集的构造与纠错码的构造有关。小偏差集是概率空间，在某种意义上近似于较大的概率空间。纠错码已经提供了这种空间的显式构造。例如，Reed-Solomon码与Hadamard码的级联提供了现在的标准构造。最近，Ben-Aroya和Ta-Shma使用Hermitian码来构造小偏差集[3]。本文考虑由扩展范数迹函数eld Fqr（x;y）=Fqr构造的小偏差集，定义为TrFqr=Fq（y）= xu，其中q是素数的幂，r2，ujqr 1 q1;这里TrFqr=Fq表示关于扩展Fqr=Fq的迹.厄米特函数eld yq + y = xq +1，它的商yq + y = xu，其中ujq + 1，以及由TrFqr=Fq（y）= NFqr=Fq（x）给出的范数迹函数eld是扩展范数迹函数eld的特殊情况。我们详细说明了由此产生的小偏差集。
As demonstrated by Naor and Naor [9] among others [1, 2], the construction of small-bias probability spaces, or small-bias sets, is connected to that of error-correcting codes. Small-bias sets are probability spaces that in some sense approximate larger ones. Error-correcting codes have provided explicit constructions of such spaces. For instance, the concatenation of a Reed-Solomon code with a Hadamard code provides a now standard construction. Recently, Ben-Aroya and Ta-Shma used Hermitian codes to construct small-bias sets [3]. In this paper, we consider small-bias sets constructed from the extended norm-trace function eld Fqr (x;y)=Fqr dened by TrFqr=Fq (y) = x u where q is a power of a prime, r 2, and uj q r 1 q 1 ; here, TrFqr=Fq denotes the trace with respect to the extension Fqr=Fq. The Hermitian function eld y q + y = x q+1 , its quotient y q + y = x u where ujq + 1, and the norm-trace function eld given by TrFqr=Fq (y) = NFqr=Fq (x) are special cases of the extended norm-trace function eld. We detail the resulting small-bias sets.