On the Failing Cases of the Johnson Bound for Error-Correcting Codes

A central problem in coding theory is to determine Aq(n, 2e + 1), the maximal cardinality of a q-ary code of length n correcting up to e errors. When e is fixed and n is large, the best upper bound for A(n, 2e+1) (the binary case) is the well-known Johnson bound from 1962. This however simply reduces to the sphere-packing bound if a Steiner system S(e + 1, 2e + 1, n) exists. Despite the fact that no such system is known whenever e ≥ 5, they possibly exist for a set of values for n with positive density. Therefore in these cases no non-trivial numerical upper bounds for A(n, 2e + 1) are known. In this paper the author presents a technique for upper-bounding Aq(n, 2e + 1), which closes this gap in coding theory. The author extends his earlier work on the system of linear inequalities satisfied by the number of elements of certain codes lying in k-dimensional subspaces of the Hamming Space. The method suffices to give the first proof, that the difference between the sphere-packing bound and Aq(n, 2e + 1) approaches infinity with increasing n whenever q and e ≥ 2 are fixed. A similar result holds for Kq(n,R), the minimal cardinality of a q-ary code of length n and covering radius R. Moreover the author presents a new bound for A(n, 3) giving for instance A(19, 3) ≤ 26168.