Linearized Reed-Solomon (LRS) codes are evaluation codes based on skew polynomials. They achieve the Singleton bound in the sum-rank metric and therefore are known as maximum sum-rank distance (MSRD) codes. In this work, we give necessary and sufficient conditions for the existence of MSRD codes with a support-constrained generator matrix. The conditions on the support constraints are identical to those for MDS codes and MRD codes. The required field size for an$[n,k]_{q^{m}}$LRS codes with support-constrained generator matrix is$q\geq \ell +1$and$m\geq \max _{l\in [\ell]}\{k-1+\log _{q}k, n_{l}\}$, where$\ell $is the number of blocks and$n_{l}$is the size of the l-th block. The special cases of the result coincide with the known results for Reed-Solomon codes and Gabidulin codes. For the support constraints that do not satisfy the necessary conditions, we derive the maximum sum-rank distance of a code whose generator matrix fulfills the constraints. Such a code can be constructed from a subcode of an LRS code with a sufficiently large field size. Moreover, as an application in network coding, the conditions can be used as constraints in an integer programming problem to design distributed LRS codes for a distributed multi-source network.

This paper considers data secrecy in distributed storage systems (DSSs) using maximally recoverable locally repairable codes (MR-LRCs). Conventional MR-LRCs are in general not secure against eavesdroppers who can observe the transmitted data during a global repair operation. This work enables nonzero secrecy dimension of DSSs encoded by MR-LRCs through a new repair framework. The key idea is to associate each local group with a central processing unit (CPU), which aggregates and transmits the contribution from the intact nodes of their group to the CPU of a group needing a global repair. The aggregation is enabled by so-called local polynomials that can be generated independently in each group. Two different schemes - direct repair and forwarded repair - are considered, and their secrecy dimension using MR-LRCs is derived. Positive secrecy dimension is enabled for several parameter regimes.

Linearized Reed-Solomon (LRS) codes are a class of evaluation codes based on skew polynomials. They achieve the Singleton bound in the sum-rank metric, and therefore are known as maximum sum-rank distance (MSRD) codes. In this work, we give necessary and sufficient conditions on the existence of MSRD codes with support-constrained generator matrix. These conditions are identical to those for MDS codes and MRD codes. Moreover, the required field size for an ${\left[ {n,k} \right]_{{q^m}}}$ LRS codes with support-constrained generator matrix is q⩾ ℓ + 1 and m ⩾ maxl∈[ℓ]{k−1+logqk,nl}, where ℓ is the number of blocks and nlis the size of the l-th block. The special cases of the result coincide with the known results for Reed-Solomon codes and Gabidulin codes.

This paper is concerned with list decoding of 2-interleaved binary alternant codes. The principle of the proposed algorithm is based on a combination of a list decoding algorithm for (interleaved) Reed-Solomon codes and an algorithm for (non-interleaved) alternant codes. A new upper bound on the decoding radius is derived and the list size is shown to scale polynomially in the code parameters. While it remains an open problem whether this upper bound is achievable, the provided simulation results show that a decoding radius exceeding the binary Johnson radius can be achieved with a high probability of decoding success by the proposed algorithm.

Generalized Goppa codes are defined by a code locator set$L$of polynomials and a Goppa polynomial G(x). When the degree of all code locator polynomials in$L$is one, generalized Goppa codes are classical Goppa codes. In this work, binary generalized Goppa codes are investigated. First, a parity-check matrix for these codes with code locators of any degree is derived. A careful selection of the code locators leads to a lower bound on the minimum Hamming distance of generalized Goppa codes which improves upon previously known bounds. A quadratic-time decoding algorithm is presented which can decode errors up to half of the minimum distance. Interleaved generalized Goppa codes are introduced and a joint decoding algorithm is presented which can decode errors beyond half the minimum distance with high probability. Finally, some code parameters and how they apply to the Classic McEliece post-quantum cryptosystem are shown.

Interleaved Reed–Solomon codes admit efficient decoding algorithms which correct burst errors far beyond half the minimum distance in the random errors regime, e.g., by computing a common solution to the Key Equation for each Reed–Solomon code, as described by Schmidt et al. If this decoder does not succeed, it may eitherfailto return a codeword ormiscorrectto an incorrect codeword, and good upper bounds on the fraction of error matrices for which these events occur are known. The decoding algorithm immediately applies to interleaved alternant codes as well, i.e., the subfield subcodes of interleaved Reed–Solomon codes, but the fraction of decodable error matrices differs, since the error is now restricted to a subfield. In this paper, we present new general lower and upper bounds on the fraction of error matrices decodable by Schmidt et al.’s decoding algorithm, thereby making it the only decoding algorithm for interleaved alternant codes for which such bounds are known.

We study scalar-linear and vector-linear solutions of the generalized combination network. We derive new upper and lower bounds on the maximum number of nodes in the middle layer, depending on the network parameters and the alphabet size. These bounds improve and extend the parameter range of known bounds. Using these new bounds we present a lower bound and an upper bound on the gap in the alphabet size between optimal scalar-linear and optimal vector-linear network coding solutions. For a fixed network structure, while varying the number of middle-layer nodes r, the asymptotic behavior of the upper and lower bounds shows that the gap is in Θ(log(r)).

We study scalar-linear and vector-linear solutions to the generalized combination network. We derive new upper and lower bounds on the maximum number of nodes in the middle layer, depending on the network parameters. These bounds improve and extend the parameter range of known bounds. Using these new bounds we present a general lower bound on the gap in the alphabet size between scalar-linear and vector-linear solutions.

Interleaved Reed–Solomon codes admit efficient decoding algorithms which correct burst errors far beyond half the minimum distance in the random errors regime, e.g., by computing a common solution to the Key Equation for each Reed–Solomon code, as described by Schmidt et al. If this decoder does not succeed, it may either fail to return a codeword or miscorrect to an incorrect codeword, and good upper bounds on the fraction of error matrices for which these events occur are known. The decoding algorithm immediately applies to interleaved alternant codes as well, i.e., the subfield subcodes of interleaved Reed–Solomon codes, but the fraction of decodable error matrices differs, since the error is now restricted to a subfield. In this paper, we present new general lower and upper bounds on the fraction of decodable error matrices by Schmidt et al.’s decoding algorithm, thereby making it the only decoding algorithm for interleaved alternant codes for which such bounds are known.

Goppa Codes are a well-known class of codes with, among others, applications in code-based cryptography. In this paper, we present a collaborative decoding algorithm for interleaved Goppa codes (IGC). Collaborative decoding increases the decoding radius beyond half of the designed minimum distance. We consider wild Goppa codes and show that we can collaboratively correct more errors for binary Goppa codes than the Patterson decoder. We propose a modified version of the McEliece cryptosystem using wild IGC based on a recently proposed system by Elleuch et al., analyze attacks on the system and present some parameters with the corresponding key sizes.