Fast Decoding of Interleaved Linearized Reed-Solomon Codes and Variants

Abstract

We construct s-interleaved linearized Reed-Solomon (ILRS) codes and variants and propose efficient decoding schemes that can correct errors beyond the unique decoding radius in the sum-rank, sum-subspace and skew metric. The proposed interpolationbased scheme for ILRS codes can be used as a list decoder or as a probabilistic unique decoder that corrects errors of sum-rank up to t ≤ s s+1 (n − k), where s is the interleaving order, n the length and k the dimension of the code. Upper bounds on the list size and the decoding failure probability are given where the latter is based on a novel Loidreau-Overbeck-like decoder for ILRS codes. The results are extended to decoding of lifted interleaved linearized Reed-Solomon (LILRS) codes in the sum-subspace metric and interleaved skew Reed-Solomon (ISRS) codes in the skew metric.We generalize fast minimal approximant basis interpolation techniques to obtain efficient decoding schemes for ILRS codes (and variants) with subquadratic complexity in the code length.Up to our knowledge, the presented decoding schemes are the first being able to correct errors beyond the unique decoding region in the sum-rank, sum-subspace and skew metric. The results for the proposed decoding schemes are validated via Monte Carlo simulations.

Decoding of Interleaved Linearized Reed-Solomon Codes with Applications to Network Coding

“…In this paper, we define and analyze LILRS codes that are obtained by lifting interleaved linearized Reed-Solomon (ILRS) codes as considered in [31]. We propose and analyze two decoding schemes that both allow for decoding insertions and deletions beyond the unique decoding region by allowing a (potentially exponential) list or a small decoding failure probability.…”

“…The element-wise application of the operator to matrices does not affect the rank, i.e. we have that rk q m (D j a (X)) = rk q m (X) (see [31,Lemma 3]). Definition 4 (σ-Generalized Moore Matrix).…”

“…In this section we consider LILRS codes for transmission over a multishot operator channel (3). We generalize the ideas from [4] to obtain multishot subspace codes by lifting the ILRS codes defined in [31]. Definition 6 (Lifted Interleaved Linearized Reed-Solomon Code).…”

“…The concept of the Loidreau-Overbeck decoder was generalized to decoding ILRS codes in the sum-rank metric [31]. In the sum-rank-metric case an F q -linear transformation matrix is obtained for each block.…”

“…Problem 1 can be solved by the skew Kötter interpolation [48] with the generalized operator evaluation maps E (i) j as defined in (15) requiring O s 2 n 2 r operations in F q m . A solution of Problem 1 can be found efficiently requiring only O(s ω M(n r )) operations in F q m using a variant of the minimal approximant bases approach from [31]. Another approach yielding the same computational complexity of O(s ω M(n r )) operations in F q m is given by the fast divide-and-conquer Kötter interpolation from [49].…”