The Covering Radius of Hadamard Codes in Odd Graphs

Sol& P., A. Ghafoor and S.A. Sheikh, The covering radius of Hadamard codes in odd graphs, Discrete Applied Mathe-atics 37/38 (1992) 501-5 10. The use of odd graphs has been proposed as fault-tolerant interconnection networks. The following problem originated in their design: what is the graphical covering radius of an Hadamard code of length 2k1 and siLe 2k1 in the odd graph Ok? Of particular interest is the case of k =2”‘‘, where we can choose this Hadamard code to be a subcode of the punctured first order Reed-Muller code RM( 1, rn). We define the w-covc :ing radius of a binary code as the largest Hamming distance from a binary word of Hamming weight M’ to the code. The above problem amounts to finding the k-covering radius of a (2k,4k, k1) Hadamard code. We find upper and lower bounds on this integer, and determine it for small values of k. Our study suggests a new isomorphism test for Hadamard designs.