On asymptotics of certain sums arising in coding theory

T. Klove (see ibid., vol.41, p.298-300, 1995) analyzed the average worst case probability of undetected error for linear [n, k; q] codes of length n and dimension k over an alphabet of size q. The following sum: S/sub n/=/spl Sigma//sub i=1//sup n/(/sub i//sup n/)(/sup i///sub n/)/sup i/((1-i)/n)/sup n-i/ arose, which also has applications in coding theory, average case analysis of algorithms, and combinatorics. Klove conjectured an asymptotic expansion of this sum, and we prove its enhanced version. Furthermore, we consider a more challenging sum arising in the upper bound of the average worst case probability of undetected error over systematic codes derived by Massey (1978). Namely S/sub n,k/=/spl Sigma//sub i=1//sup n/(/sub i//sup n-k/)(/sup i///sub n/)/sup i/((1-i)/n)/sup n-i/ for k/spl ges/0. We obtain an asymptotic expansion of S/sub n,k/, and this leads to a conclusion that Massey's bound on the average worst case probability over all systematic codes is better for every k than the corresponding Klove's bound over all codes [n, k; q]. The technique used belongs to the analytical analysis of algorithms and is based on some enumeration of trees, singularity analysis, Lagrange's inversion formula, and Ramanujan's identities. In fact, S/sub n/, turns out to be related to the so-called Ramanujan's Q-function which finds many applications (e.g. hashing with linear probing, the birthday paradox problem, random mappings, caching, memory conflicts, etc.).