Decoder error probability of bounded distance decoders for constant-dimension codes

Constant-dimension codes (CDCs) have been considered for error correction in random linear network coding, and low-complexity bounded distance decoders have been proposed. However, error performance, decoder error probability (DEP) in particular, of these bounded distance decoders has received little attention. In this paper, we first establish some fundamental geometric properties of the projective space. In particular, we show that the volume of the intersection of two spheres depends on only the two radii as well as the distance between and the dimensions of the two centers. Using these geometric properties, we then consider bounded distance decoders in both subspace and injection metrics and derive analytical expressions of their DEPs for CDCs over a symmetric operator channel, which ultimately depend on their distance distributions. Finally, we focus on CDCs obtained by lifting rank metric codes since their distance distributions are known, and obtain two important results. First, we obtain asymptotically tight upper bounds on the DEPs of bounded distance decoders in both metrics; the upper bounds decrease exponentially with the square of the minimum distance. Second, we show that the DEP for KK codes, obtained by lifting Gabidulin codes, is the highest up to a scalar among all CDCs obtained by lifting rank metric codes.