LCD subspace codes

TL;DR

This paper introduces LCD subspace codes, explores their decoding advantages, and presents constructions using association schemes and mutually unbiased matrices, expanding the theoretical framework of subspace coding.

Contribution

It defines LCD subspace codes, demonstrates their simplified decoding, and provides new construction methods using association schemes and Hadamard matrices.

Findings

Decoding for LCD subspace codes is simpler than for general subspace codes.

Equitable partitions of association schemes can produce LCD subspace codes.

Constructs of LCD subspace codes from mutually unbiased Hadamard and weighing matrices.

Abstract

A subspace code is a nonempty set of subspaces of a vector space $\mathbb F^n_q$. Linear codes with complementary duals, or LCD codes, are linear codes whose intersection with their duals is trivial. In this paper, we introduce a notion of LCD subspace codes. We show that the minimum distance decoding problem for an LCD subspace code reduces to a problem that is simpler than for a general subspace code. Further, we show that under some conditions equitable partitions of association schemes yield such LCD subspace codes and as an illustration of the method give some examples from distance-regular graphs. We also give a construction from mutually unbiased Hadamard matrices, and more generally, from mutually unbiased weighing matrices.