The sum-rank metric generalizes the Hamming and rank metric by partitioning vectors into blocks and defining the total weight as the sum of the rank weights of these blocks, based on their matrix representation. In this work, we explore support-guessing algorithms for decoding sum-rank-metric codes. Support-guessing involves randomly selecting candidate supports and attempting to decode the error under the assumption that it is confined to these supports. While previous works have focused on worst-case scenarios, we analyze the average case and derive an optimal support-guessing distribution in the asymptotic regime. We show that this distribution also performs well for finite code lengths. Our analysis provides exact complexity estimates for unique decoding scenarios and establishes tighter bounds beyond the unique decoding radius. Additionally, we introduce a randomized decoding algorithm for linearized Reed–Solomon codes. This algorithm extends decoding capabilities beyond the unique decoding radius by leveraging an efficient error-and-erasure decoder. Instead of requiring the entire error support to be confined to the guessed support, the algorithm succeeds as long as there is sufficient overlap between the guessed support and the actual error support. As a result, the proposed method improves the success probability and reduces computational complexity compared to generic decoding algorithms. Our contributions offer more accurate complexity estimates than previous works and clarify how support-guessing methods behave under different assumptions on the code class, rank profile, and decoding radius. These estimates provide a codingtheoretic basis for future work on decoding algorithms and applications of the sum-rank metric.

Abstract This paper provides new bounds on the size of spheres in any coordinate-additive metric with a particular focus on improving existing bounds in the sum-rank metric. We derive improved upper and lower bounds based on the entropy of a distribution related to the Boltzmann distribution, which work for any coordinate-additive metric. Additionally, we derive new closed-form upper and lower bounds specifically for the sum-rank metric that outperform existing closed-form bounds.

Abstract Skew polynomials are a class of non-commutative polynomials that have several applications in computer science, coding theory and cryptography. In particular, skew polynomials can be used to construct and decode evaluation codes in several metrics, like e.g. the Hamming, rank, sum-rank and skew metric. We propose a fast divide-and-conquer variant of Kötter–Nielsen–Høholdt (KNH) interpolation algorithm: it inputs a list of linear functionals on skew polynomial vectors, and outputs a reduced Gröbner basis of their kernel intersection. We show, that the proposed KNH interpolation can be used to solve the interpolation step of interpolation-based decoding of interleaved Gabidulin codes in the rank-metric, linearized Reed–Solomon codes in the sum-rank metric and skew Reed–Solomon codes in the skew metric requiring at most $${\widetilde{O}}\left( s^\omega {\mathfrak {M}}(n)\right) $$ O ~ s ω M ( n ) operations in $${\mathbb {F}}_{q^m}$$ F q m , where n is the length of the code, $$s$$ s the interleaving order, $${\mathfrak {M}}(n)$$ M ( n ) the complexity for multiplying two skew polynomials of degree at most n, $$\omega $$ ω the matrix multiplication exponent and $${\widetilde{O}}\left( \cdot \right) $$ O ~ · the soft-O notation which neglects log factors. This matches the previous best speeds for these tasks, which were obtained by top–down minimal approximant bases techniques, and complements the theory of efficient interpolation over free skew polynomial modules by the bottom-up KNH approach. In contrast to the top–down approach the bottom-up KNH algorithm has no requirements on the interpolation points and thus does not require any pre-processing.

In this paper we address the problem of decoding linearized Reed–Solomon (LRS) codes beyond their unique decoding radius. We analyze the complexity in order to evaluate if the considered problem is of cryptographic relevance, i.e., can be used to design cryptosystems that are computationally hard to break. We show that our proposed algorithm improves over other generic algorithms that do not take into account the underlying code structure.

This paper investigates the decoding of certain Gabidulin codes over a channel with space-symmetric errors. Space-symmetric errors are additive error matrices that have the property that their column and row spaces are equal. We show that for channels restricted to space-symmetric errors, with high probability errors of rank up to$2 (n-k)/3$can be decoded with a Gabidulin code of length$n$and dimension$k$, using a weak-self orthogonal basis as code locators.

We speed up existing decoding algorithms for three code classes in different metrics: interleaved Gabidulin codes in the rank metric, lifted interleaved Gabidulin codes in the subspace metric, and linearized Reed-Solomon codes in the sum-rank metric. The speed-ups are achieved by new algorithms that reduce the cores of the underlying computational problems of the decoders to one common tool: computing left and right approximant bases of matrices over skew polynomial rings. To accomplish this, we describe a skew-analogue of the existing PM-Basis algorithm for matrices over ordinary polynomials. This captures the bulk of the work in multiplication of skew polynomials, and the complexity benefit comes from existing algorithms performing this faster than in classical quadratic complexity. The new algorithms for the various decoding-related computational problems are interesting in their own and have further applications, in particular parts of decoders of several other codes and foundational problems related to the remainder-evaluation of skew polynomials.

The key agreement problem with biometric or physical identifiers and two terminals for key enrollment and reconstruction is considered. A nested convolutional code construction that performs lossy compression with side information is proposed. Nested convolutional codes are an alternative to nested polar codes and nested random linear codes that achieve all points of the key-leakage-storage regions of the generated-secret and chosen-secret models for long block lengths. Our design uses a convolutional code for vector quantization during enrollment and a subcode of it for error correction during reconstruction. Physical identifiers with small bit error probability are considered to illustrate the gains of the proposed construction. One variant of nested convolutional codes improves on all previous constructions in terms of the key vs. storage rate ratio but it has high complexity. Another variant of nested convolutional codes with lower complexity performs similarly to previously designed nested polar codes. The results suggest that the choice of convolutional or polar codes for key agreement with identifiers depends on the complexity constraints.

We show that the root-finding step in interpolation-based decoding of interleaved Gabidulin codes can be solved by finding a so-called minimal approximant basis of a matrix over a linearized polynomial ring. Based on existing fast algorithms for computing such bases over ordinary polynomial rings, we develop fast algorithms for computing them over linearized polynomials. As a result, root finding costs O~(ℓωM(n)) operations in Fqm, where ℓ is the interleaving degree, n the code length, Fqm the base field of the code, 2 ≤ ω ≤ 3 the matrix multiplication exponent, and M(n) ∈ O(n1.635) is the complexity of multiplying two linearized polynomials of degree at most n. This is an asymptotic improvement upon the previously fastest algorithm of complexity O(ℓ3n2), in some cases O(ℓ2n2).

In this paper, we provide an analysis of the performance of concatenation of polar codes with outer cyclic redundancy check (CRC) codes, separated by an interleaver, in the short and moderate block length regimes. The analysis addresses maximum likelihood decoding as a proxy to the code performance under successive cancellation list decoding. The analysis is carried out by introducing the concatenated polar code (CPC) ensembles, whose distance properties can be analyzed (for sufficiently short block lengths) by means of the uniform interleaver approach. At moderate block lengths, we resort to the Monte Carlo simulations. Results show that if the inner polar code possesses a low minimum distance and the outer CRC code has a sufficiently large amount of redundancy, then the choice of the outer code generator polynomial and the interleaver may yield to a large variability in the performance of the resulting CPC.