基于代数函数域的子空间设计
Subspace Designs based on Algebraic Function Fields
摘要：
子空间设计是$\F_q^m$的高维子空间$\{H_i\}$的（大）集合，使得对于任何低维子空间$W$，集合中只有少数子空间与$W$有非平凡交集;更确切地说，$W \cap H_i$的维数之和至多是某个参数$L$。这个概念是由Guruswami和Xing（STOC'13）提出的，应用于Reed-Solomon和代数几何码的列表解码变体，后来也用于具有最佳列表解码半径的显式秩度量码。Guruswami和Kopparty（FOCS'13，Combinatorica'16）给出了具有近优参数的子空间设计的显式构造。这种构造基于多项式，与折叠里德-所罗门码有密切联系，并且需要大的字段大小（特别是$q \ge m$）。Forbes和Guruswami（RANDOM'15）使用这种构造在大的域上给出了显式的常次数“维数扩展器”，并指出子空间设计是线性代数伪随机性的有力工具。在这里，我们构建子空间设计在任何领域，在总的交叉维度上的界$L$的适度恶化为代价。我们的方法是基于一个（非平凡的）扩展的多项式为基础的建设代数函数字段，并实例化的方法与分圆函数字段。在Forbes和Guruswami的构造中插入我们的新子空间设计，得到了$\F^n$上的维数扩展器，对于任何域$\F$，具有对数度和扩展保证，对于维数为$Ω(n/(\log \log n))$的子空间。
英文摘要：
Subspace designs are a (large) collection of high-dimensional subspaces $\{H_i\}$ of $\F_q^m$ such that for any low-dimensional subspace $W$, only a small number of subspaces from the collection have non-trivial intersection with $W$; more precisely, the sum of dimensions of $W \cap H_i$ is at most some parameter $L$. The notion was put forth by Guruswami and Xing (STOC'13) with applications to list decoding variants of Reed-Solomon and algebraic-geometric codes, and later also used for explicit rank-metric codes with optimal list decoding radius. Guruswami and Kopparty (FOCS'13, Combinatorica'16) gave an explicit construction of subspace designs with near-optimal parameters. This construction was based on polynomials and has close connections to folded Reed-Solomon codes, and required large field size (specifically $q \ge m$). Forbes and Guruswami (RANDOM'15) used this construction to give explicit constant degree "dimension expanders" over large fields, and noted that subspace designs are a powerful tool in linear-algebraic pseudorandomness. Here, we construct subspace designs over any field, at the expense of a modest worsening of the bound $L$ on total intersection dimension. Our approach is based on a (non-trivial) extension of the polynomial-based construction to algebraic function fields, and instantiating the approach with cyclotomic function fields. Plugging in our new subspace designs in the construction of Forbes and Guruswami yields dimension expanders over $\F^n$ for any field $\F$, with logarithmic degree and expansion guarantee for subspaces of dimension $Ω(n/(\log \log n))$.
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