Abstract Poset metrics in the context of stabilizer codes are investigated. We establish a Singleton-type bound for stabilizer poset codes and introduce the notion of MDS stabilizer poset codes. Several characterizations of these codes are obtained, including criteria in terms of additive codes and perfect codes. We also prove existence results for large families of examples, thereby extending classical results on MDS poset codes to the quantum setting.

Abstract This article introduces Generalized Hyperderivative Reed–Solomon codes (GHRS codes), which generalize NRT Reed–Solomon codes. Its main results are as follows: (1) every GHRS code is MDS, (2) the dual of a GHRS code is also an GHRS code, (3) determine subfamilies of GHRS codes whose members are low-density parity-check codes (LDPCs), and (4) determine a family of GHRS codes whose members are quasi-cyclic. We point out that there are GHRS codes having all of these properties.

We show that it is possible to break up Higher Grassmann codes we constructed in earlier work to a sequence of affine Reed-Muller codes so that various operations such as decoding can be reduced to performing these operations for the component affine Reed-Muller codes. Then we consider quantum codes produced from a pair of Higher Grassmann codes and discuss also their encoding and decoding aspects in some detail.

This paper introduces the concept of necklace binomial coefficients motivated by the enumeration of a special type of sequences. Several properties of these coefficients are described, including a connection between their roots and an elliptic curve. Further links are given to a physical model from quantum mechanical supersymmetry as well as properties of alkane molecules in chemistry.