Certain Types of Linear Codes over the Finite Field of Order Twenty-Five

The aim of the paper is to compute projective maximum distance separable codes, -MDS of two and three dimensions with certain lengths and Hamming weight distribution from the arcs in the projective line and plane over the finite field of order twenty-five. Also, the linear codes generated by an incidence matrix of points and lines of were studied over different finite fields.

A New Class of Q-Ary Codes for the McEliece Cryptosystem

The McEliece cryptosystem is a promising candidate for post-quantum public-key encryption. In this work, we propose q-ary codes over Gaussian integers for the McEliece system and a new channel model. With this one Mannheim error channel, errors are limited to weight one. We investigate the channel capacity of this channel and discuss its relation to the McEliece system. The proposed codes are based on a simple product code construction and have a low complexity decoding algorithm. For the one Mannheim error channel, these codes achieve a higher error correction capability than maximum distance separable codes with bounded minimum distance decoding. This improves the work factor regarding decoding attacks based on information-set decoding.

Quantum Codes of Maximal Distance and Highly Entangled Subspaces

We present new bounds on the existence of general quantum maximum distance separable codes (QMDS): the length n of all QMDS codes with local dimension D and distance d≥3 is bounded by n≤D2+d−2. We obtain their weight distribution and present additional bounds that arise from Rains' shadow inequalities. Our main result can be seen as a generalization of bounds that are known for the two special cases of stabilizer QMDS codes and absolutely maximally entangled states, and confirms the quantum MDS conjecture in the special case of distance-three codes. As the existence of QMDS codes is linked to that of highly entangled subspaces (in which every vector has uniform r-body marginals) of maximal dimension, our methods directly carry over to address questions in multipartite entanglement.

New LCD MDS codes constructed from generalized Reed–Solomon codes

Maximum distance separable codes with complementary duals (LCD MDS codes) are very important in coding theory and practice, and have attracted a lot of attention. In this paper, we focus on LCD MDS codes constructed from generalized Reed–Solomon (GRS) codes over a finite field with odd characteristic. We detail two constructions of new LCD MDS codes, using invertible matrices and the roots of three classes of polynomials, respectively.

Maximum distance separable codes over ℤ2 × ℤ2s

Bilal et al. (Maximum distance separable codes over [Formula: see text] and [Formula: see text] Des. Codes Cryptogr. 61 (2011) 31–40) obtained two upper bounds on minimum distance of codes over rings to the case of [Formula: see text]-additive codes and through these bounds, they introduced two kinds of maximum distance separable codes ([Formula: see text] and [Formula: see text]), the minimum distance of which meets any of those bounds. Also, they completely determined these two types of codes. In this paper, we generalize these facts on [Formula: see text]-additive codes and determine all possible parameters of the [Formula: see text] and [Formula: see text] codes over [Formula: see text].