13 research outputs found
On the List-Decodability of Random Linear Rank-Metric Codes
The list-decodability of random linear rank-metric codes is shown to match
that of random rank-metric codes. Specifically, an -linear
rank-metric code over of rate is shown to be (with high probability)
list-decodable up to fractional radius with lists of size at
most , where is a constant
depending only on and . This matches the bound for random rank-metric
codes (up to constant factors). The proof adapts the approach of Guruswami,
H\aa stad, Kopparty (STOC 2010), who established a similar result for the
Hamming metric case, to the rank-metric setting
Subspace Designs Based on Algebraic Function Fields
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\u2713) 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\u2713, Combinatorica\u2716) 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 >= m). Forbes and Guruswami (RANDOM\u2715) 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 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 Omega(n/(log(log(n))))
Lossless Dimension Expanders via Linearized Polynomials and Subspace Designs
For a vector space F^n over a field F, an (eta,beta)-dimension expander of degree d is a collection of d linear maps Gamma_j : F^n -> F^n such that for every subspace U of F^n of dimension at most eta n, the image of U under all the maps, sum_{j=1}^d Gamma_j(U), has dimension at least beta dim(U). Over a finite field, a random collection of d = O(1) maps Gamma_j offers excellent "lossless" expansion whp: beta ~~ d for eta >= Omega(1/d). When it comes to a family of explicit constructions (for growing n), however, achieving even modest expansion factor beta = 1+epsilon with constant degree is a non-trivial goal.
We present an explicit construction of dimension expanders over finite fields based on linearized polynomials and subspace designs, drawing inspiration from recent progress on list-decoding in the rank-metric. Our approach yields the following:
- Lossless expansion over large fields; more precisely beta >= (1-epsilon)d and eta >= (1-epsilon)/d with d = O_epsilon(1), when |F| >= Omega(n).
- Optimal up to constant factors expansion over fields of arbitrarily small polynomial size; more precisely beta >= Omega(delta d) and eta >= Omega(1/(delta d)) with d=O_delta(1), when |F| >= n^{delta}. Previously, an approach reducing to monotone expanders (a form of vertex expansion that is highly non-trivial to establish) gave (Omega(1),1+Omega(1))-dimension expanders of constant degree over all fields. An approach based on "rank condensing via subspace designs" led to dimension expanders with beta >rsim sqrt{d} over large fields. Ours is the first construction to achieve lossless dimension expansion, or even expansion proportional to the degree
Bounded-degree Low Rank Parity Check Codes
Low rank parity check (LRPC) codes are the rank-metric analogue of low
density parity check codes. In this paper we investigate a sub-family of LRPC
codes, which have a parity check matrix defined over a subspace
, where
is the finite field of elements and is
significantly smaller than . These codes are named bounded-degree LRPC
(BD-LRPC) codes and are the same as the standard LRPC codes of density when
the degree , while BD-LRPC codes of degree constitute a proper
subset of LRPC codes of density . Exploiting the particular structure of
their parity check matrix, we show that the BD-LRPC codes of degree can
uniquely correct errors of rank weight when for certain , in contrast to the condition required for the standard
LRPC codes, where . This underscores the superior decoding
capability of the proposed BD-LRPC codes. As the code length approaches
infinity, when , it is shown that can be chosen as a
certain constant, which indicates that the BD-LRPC codes with a code rate of
can be, with a high probability, uniquely decodable with the decoding
radius approaching the Singleton bound for ; and
when is a constant, the BD-LRPC codes can have unique decoding radius
for a small , which can easily lead to
with properly chosen parameters.Comment: Currently under revie