Constructions and bounds for subspace codes
Sascha Kurz
TL;DR
This survey consolidates key theories and methods for subspace codes, focusing on the q-analog framework that parallels binary coding theory. It highlights the central role of rank-metric codes and lifting in constructing constant-dimension codes, and it catalogs a spectrum of upper bounds (anticode, sphere-packing, Johnson, LP/SDP) while detailing parametric bounds for partial spreads via divisible-point structures. The Constructions section organizes the main building blocks—lifting, linkage, EF, and coset constructions—and shows how d-packings and multi-component designs push the size frontier for CDCs. Collectively, the chapter maps foundational tools, interconnections, and practical strategies for advancing subspace code design and analysis, with attention to asymptotics and open problems.
Abstract
Subspace codes are the $q$-analog of binary block codes in the Hamming metric. Here the codewords are vector spaces over a finite field. They have e.g. applications in random linear network coding, distributed storage, and cryptography. In this chapter we survey known constructions and upper bounds for subspace codes.