r-Minimal Codes with Respect to Rank Metric

Abstract

In this paper, we propose and study r-minimal codes, a natural extension of minimal codes which have been extensively studied with respect to Hamming metric, rank metric and sum-rank metric. We first propose r-minimal codes in a general setting where the ambient space is a finite dimensional left module over a division ring and is supported on a lattice. We characterize minimal subcodes and r-minimal codes, derive a general singleton bound, and give existence results for r-minimal codes by using combinatorial arguments. We then consider r-minimal rank metric codes over a field extension E/F of degree m, where E can be infinite unless otherwise specified. We characterize these codes in terms of cutting r-blocking sets, generalized rank weights of the codes and those of the dual codes, and classify codes whose r-dimensional subcodes have constant rank support weight. Next, with the help of the evasiveness property of cutting r-blocking sets and some upper bounds for the dimensions of evasive subspaces, we derive several lower and upper bounds for the minimal length of r-minimal codes. Furthermore, when E is finite, we establish a general upper bound which generalizes and improves the counterpart for minimal codes in the literature. As a corollary, we show that if m = 3, then for any k ⩾ 2, the minimal length of k-dimensional minimal codes is equal to 2k. To the best of our knowledge, when m ⩾ 3, there is no known explicit formula for the minimal length of k-dimensional minimal codes for arbitrary k in the literature.