Several new lower bounds for football pool systems.
Uri Blass, Simon N. Litsyn · 1998
We derive several new lower bounds on the size of ternary covering codes of lengths 6, 7 and 8 and with covering radii 2 or 3. 1 Introduction Ternary covering code C of length n and radius R is a collection of ternary vectors of length n possessing the property that every ternary vector of length n differs from at least one codeword in at most R coordinates. We wish to minimize the size of C. The search for ternary covering codes attracted a big deal of attention due to its equivalence to constructing systems for football pools. A survey of what is known on the problem can be found in [8, 7, 11]. Along with constructing such codes, lower bounds on their size are extensively studied [2, 4, 5, 6, 9, 12, 13]. The known lower bounds give reasonably good results in cases when R is relatively small in comparison with n. In the paper we present an approach giving several better lower bounds in some situations when R is relatively big. Let K 3 (n; R) be the minimal size of a ternary code of l...