In this work, we prove that the permutation group of a Reed-Solomon code is given by the polynomials of degree one that leave the set of evaluation points invariant. Our results provide a straightforward proof of the well-known cases of the permutation group of the Reed-Solomon code when the set of evaluation points is the whole finite field or the multiplicative group.

In this paper, we consider using norm-trace codes over extension fields for error correction using partial information from received words. To do so, we define virtual projections of norm-trace codes and we implement a fractional decoding scheme. The scheme depends on a refined key equation tailored to the norm-trace code.

We consider the problem of secure distributed matrix multiplication (SDMM), where a user has two matrices and wishes to compute their product with the help of N honest but curious servers under the security constraint that any information about either A or B is not leaked to any server. This paper presents a new scheme that considers the inner product partition for matrices A and B. Our central technique relies on encoding matrices A and B in a Hermitian code and its dual code, respectively. We present the Hermitian Algebraic (HerA) scheme, which employs Hermitian codes and characterizes the partitioning and security capacities given entries of matrices belonging to a finite field with q2elements. We showcase that this scheme performs the secure distributed matrix multiplication in a significantly smaller finite field and expands security allowances compared to the existing results in the literature.

We present a new probabilistic decoding algorithm that can be used to perform fractional decoding of codes from the Hermitian curve. Fractional decoding means that the original codeword may be obtained from a received word using only an$\alpha$-proportion of symbols of the received word, provided not too many errors have occurred. The procedure presented makes use of fractional decoding of Reed-Solomon codes while allowing for the use of codes of similar lengths over smaller fields.

In this paper we consider self-dual NRT codes, that is, self-dual codes in the metric space endowed with the Niederreiter-Rosenbloom-Tsfasman metric (NRT metric) and their shape enumerators as defined by Barg and Park. We use polynomial invariant theory to describe the shape enumerator of a binary self-dual NRT code, even self-dual NRT code, and weak doubly even self-dual NRT code in Mn,2(F2). Motivated by these results, we describe the number of invariant polynomials that we must find to describe the shape enumerator of a self-dual NRT code in Mn,s(F2). We define the ordered flip of a matrix A ∈ Mk,ns(Fq) and present some constructions of self-dual NRT codes over Fq. We further give an application of ordered flip to the classification of self-dual NRT codes of dimension two.

We define a virtual projection of a Reed-Solomon code RS(ql, n, k) to an RS(q, n, k) Reed-Solomon code. A new probabilistic decoding algorithm that can be used to perform fractional decoding beyond the α- decoding radius is considered. An upper bound for the failure probability of the new algorithm is given, and the performance is illustrated by examples.