Codes in the projective space and codes in the Grassmannian over a finitefield - referred to as subspace codes and constant-dimension codes (CDCs),respectively - have been proposed for error control in random linear networkcoding. For subspace codes and CDCs, a subspace metric was introduced tocorrect both errors and erasures, and an injection metric was proposed tocorrect adversarial errors. In this paper, we investigate the packing andcovering properties of subspace codes with both metrics. We first determinesome fundamental geometric properties of the projective space with bothmetrics. Using these properties, we then derive bounds on the cardinalities ofpacking and covering subspace codes, and determine the asymptotic rates ofoptimal packing and optimal covering subspace codes with both metrics. Ourresults not only provide guiding principles for the code design for errorcontrol in random linear network coding, but also illustrate the differencebetween the two metrics from a geometric perspective. In particular, ourresults show that optimal packing CDCs are optimal packing subspace codes up toa scalar for both metrics if and only if their dimension is half of theirlength (up to rounding). In this case, CDCs suffer from only limited rate lossas opposed to subspace codes with the same minimum distance. We also show thatoptimal covering CDCs can be used to construct asymptotically optimal coveringsubspace codes with the injection metric only.
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