Cyclonomial number systems and the ranking of lexicographically ordered constant-sum codes (Q2715949)

The cyclonomial coefficient \(\binom nk_q\) is defined as the number of integer words \(w_{n-1}\dots w_0\) with \(\sum w_i =k\), \(0 \leq w_i \leq q-1\). They appear as coefficients of \((1+x+\dots + x^{q-1})^n\) and each natural number can be written, for a fixed \(k \geq 1\) and a fixed \(q \geq 2\), in a unique way as \(\binom {b_k}k_q+ \dots + \binom {b_1}1_q\), where \(b_k \geq \dots \geq b_1 \geq 0\) and where the number of consecutive \(b_i\) that are mutually equal is limited by \(q-1\). This is stated as Theorem 3.1 and then used for the index lexicographical ordering of the words \(w_{n-1} \dots w_0\). Unfortunately, there are two typos in the formula of Theorem 3.1, both at the position of indices \(q\).