26 research outputs found
Rate-distance tradeoff for codes above graph capacity
The capacity of a graph is defined as the rate of exponential growth of
independent sets in the strong powers of the graph. In the strong power an edge
connects two sequences if at each position their letters are equal or adjacent.
We consider a variation of the problem where edges in the power graphs are
removed between sequences which differ in more than a fraction of
coordinates. The proposed generalization can be interpreted as the problem of
determining the highest rate of zero undetected-error communication over a link
with adversarial noise, where only a fraction of symbols can be
perturbed and only some substitutions are allowed.
We derive lower bounds on achievable rates by combining graph homomorphisms
with a graph-theoretic generalization of the Gilbert-Varshamov bound. We then
give an upper bound, based on Delsarte's linear programming approach, which
combines Lov\'asz' theta function with the construction used by McEliece et al.
for bounding the minimum distance of codes in Hamming spaces.Comment: 5 pages. Presented at 2016 IEEE International Symposium on
Information Theor
Searching for Minimum Storage Regenerating Codes
Regenerating codes allow distributed storage systems to recover from the loss
of a storage node while transmitting the minimum possible amount of data across
the network. We present a systematic computer search for optimal systematic
regenerating codes. To search the space of potential codes, we reduce the
potential search space in several ways. We impose an additional symmetry
condition on codes that we consider. We specify codes in a simple alternative
way, using additional recovered coefficients rather than transmission
coefficients and place codes into equivalence classes to avoid redundant
checking. Our main finding is a few optimal systematic minimum storage
regenerating codes for and , over several finite fields. No such
codes were previously known and the matching of the information theoretic
cut-set bound was an open problem