## On the growth rate of convolutional codes

#### Abstract

Let $\mathcal C$ be a rate $(n-r)/n$ convolutional code with canonical parity-check matrix $\mathbf{H}(D)$, and let $w_0$ denote the minimum average weight per edge over all nonzero cycles in the state diagram for $\mathcal C$. The rate of growth per edge of the minimum distance between two nonmerged codewords is given by the value of $w_0$. For any code $\mathcal C$, we show how to obtain an upper bound on $w_0$ from the columns of $\mathbf{H}(D)$.