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Given full-rank parity-check matrices $H_\mathcal{A}$ and
$H_\mathcal{B}$ for linear binary codes $\mathcal{A}$ and
$\mathcal{B}$, respectively, two full-rank parity-check matrices,
denoted $H_{1}$ and $H_{2}$, are given for the product code
$\mathcal{A}\otimes\mathcal{B}$. It is shown that the girth of
Tanner graph $TG(H_{i})$ associated with $H_{i}, i = 1,2$, is
bounded below by $\{g_{a}, g_{b}, 8\}$ where $g_{a}$ and $g_{b}$
are the girths of $TG(H_\mathcal{A})$ and $TG(H_\mathcal{B})$,
respectively. It turns out that the product of $m \geq 2$ single
parity-check codes is either cycle-free or has girth 8, and a
necessary and sufficient condition for having the latter case is
provided.
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