centralizer algebra

Schrijver’s SDP Bound for Network Codes

This is a small report on a failed project — obtaining semidefinite programming bounds on constant dimension network codes. But let us start with some context …

A network code consists of a set of subspaces in \mathbb{F}_q^n. It is a code, so we want to maximize the distance between subspaces (or increase the code’s cardinality or rate). The most reasonable metric for this is the subspace metric in which two subspaces U and W have distance \dim(U+W) - \dim(U \cap W), that is the shortest distance between U and W in the subspace lattice. These codes are used in random linear network coding, that is a method for the many-to-many transmission of data in a network which is faster than multiple one-to-one connections. As in classical coding theory, one area of research on those subspace codes is concerned with bounding the maximal size of a code with minimum distance d in \mathbb{F}_q^n. Notice that one can interpret q=1 as a field with one element as is explained here by Peter Cameron. Then it corresponds to classical coding theory. (more…)

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