Upper bound on the minimum distance of LDPC codes over $GF(q)$ based on counting the number of syndromes

In [1] a syndrome counting based upper bound on the minimum distance of regular binary LDPC codes is given. In this paper we extend the bound to the case of irregular and generalized LDPC codes over $GF(q)$. A comparison with the lower bound for LDPC codes over $GF(q)$, upper bound for the codes over $GF(q)$, and the shortening upper bound for LDPC codes is made. The new bound is shown to lie under the Gilbert–Varshamov bound at high rates.