Explicit Subspace Designs

Abstract

A subspace design is a collection {H 1 , H 2 , . . . , H M } of subspaces of F m q with the property that no low-dimensional subspace W of F m q intersects too many subspaces of the collection. Subspace designs were introduced by Guruswami and Xing (STOC 2013) who used them to give a randomized construction of optimal rate list-decodable codes over constant-sized large alphabets and sub-logarithmic (and even smaller) list size. Subspace designs are the only non-explicit part of their construction. In this paper, we give explicit constructions of subspace designs with parameters close to the probabilistic construction, and this implies the first deterministic polynomial time construction of list-decodable codes achieving the above parameters.Our constructions of subspace designs are natural and easily described, and are based on univariate polynomials over finite fields. Curiously, the constructions are very closely related to certain good listdecodable codes (folded RS codes and univariate multiplicity codes). The proof of the subspace design property uses the polynomial method (with multiplicities): Given a target low-dimensional subspace W , we construct a nonzero low-degree polynomial P W that has several roots for each H i that non-trivially intersects W . The construction of P W is based on the classical Wronskian determinant and the folded Wronskian determinant, the latter being a recently studied notion that we make explicit in this paper. Our analysis reveals some new phenomena about the zeroes of univariate polynomials, namely that polynomials with many structured roots or many high multiplicity roots tend to be linearly independent.

Improved Decoding of Folded Reed-Solomon and Multiplicity Codes

“…The next theorem is a variant of Theorem 14 from [GK16b] for (X q , d)-closed subspaces of polynomials of degree d > q.…”

“…Since then there has been a great deal of work aimed at reducing the list size and alphabet size of these constructions, both of which were polynomial in n (and both of which would ideally be independent of n). To reduce the alphabet size to constant, two high-level strategies are known to work: (1) swapping out the standard polynomial codes for Algebraic Geometry (AG) codes [GX12, GX13,GK16b], and (2) concatenation and distance amplification using expander graphs [AEL95, GI04, HW15, GKO + 17, HRW17]. To reduce the listsize to constant, the known strategies involve passing to carefully constructing subcodes of Folded Reed-Solomon codes and univariate multiplicity codes, via pseudorandom objects such as subspace evasive sets or subspace designs [DL12, GW13, GX12, GX13,GK16b].…”

“…• Second, use Theorem 17 from [GK16b] instead of Theorem 14 from that paper, to show that for a low dimensional subspace W , at a typical a ∈ D we have dim({P (X) ∈ W | P (<s) (a) = 0}) is small.…”

“…The second step uses the d < char(F q ) condition more essentially. By a reworking of several algebraic tools used in the proof of Theorem 17 from [GK16b], we prove a generalization of it to handle polynomials of degree > q. This generalization will only apply to subspaces W of a special kind ("(X q , d)-closed" subspaces).…”

Linear-time list recovery of high-rate expander codes

“…An empty "recovery time" field means that there are no known efficient algorithms. We remark that [GX13], along with the explicit subspace designs of [GK13], also give explicit constructions of high-rate AG subcodes with polynomial time list-recovery and somewhat complicated parameters; the list-size L becomes super-constant.…”