Algorithm 1 gAD-GDBFwM with Finite State Machine in one component decoder

Input:  $\mathbf{y}=(y_i,y_2,\dots ,y_N)\in {\mathbb{R}}^N$,   ${\mathbf{r}}_{in}^{\left(t\right)}$,   ${\mathit{\mu }}_{in}^{\left(t\right)}$,   ${\mathbf{P}}_{in}^{\left(t\right)}$,   $\delta$ Output:  $\widehat{\mathbf{x}}=({\widehat{x}}_1^{(\ell )},{\widehat{x}}_2^{(\ell )},\dots ,{\widehat{x}}_N^{(\ell )})\in {\{\pm 1\}}^N$,   ${\mathbf{r}}_{out}^{\left(t\right)}$,   ${\mathit{\mu }}_{out}^{\left(t\right)}$,   ${\mathbf{P}}_{out}^{\left(t\right)}$

Initialization:  $\ell =1$, $r^{\left(1\right)}={\mathbf{r}}_{in}^{\left(t\right)}$, $\mathit{\mu }={\mathit{\mu }}_{in}^{\left(t\right)}$, $\mathbf{P}={\mathbf{P}}_{in}^{\left(t\right)}$

${\widehat{x}}_i^{\left(1\right)}=\mathrm{sign}\left(r_i^{\left(1\right)}\right)$, $i=1,2,\dots ,N$ $s_j^{\left(1\right)}={\prod }_{i\in \mathcal{Q}\left(c_j\right)}{\widehat{x}}_i^{\left(1\right)}$, $j=1,2,\dots ,M$ while ($\ell \le L_{\max ,\mathrm{t}})$ or ($s_j^{(\ell )}=1,\forall j$) do    $E_{min}^{(\ell )}=+\infty$    for $i=1,\dots ,N$ do      $E_i^{(\ell )}=w_1^{(\ell )}{y}_i{\widehat{x}}_i^{(\ell )}+w_2^{(\ell )}{\sum }_{j\in \mathcal{P}\left(v_i\right)}{s}_j^{(\ell )}+m_{{\mu }_i}$      $E_{min}^{(\ell )}=min\{E_{min}^{(\ell )},E_i^{(\ell )}\}$    end for    ${\mathcal{F}}^{(\ell )}=\left\{v_i\right|E_i\le E_{min}+\delta \}$     update $\mu$ accordingly to ${\mathcal{F}}^{(\ell )}$     update $\mathbf{P}$ accordingly to ${\mathcal{F}}^{(\ell )}$ and state machine    $r_i^{(\ell +1)}=r_i^{(\ell )}+s_{v_i}\mathrm{sign}\left(r_i^{(\ell )}\right)$, $i=1,2,\dots ,N$     apply rule $\theta$ if necessary and update $r_i^{(\ell +1)}$, $i=1,2,\dots ,N$    ${\widehat{x}}_i^{(\ell +1)}=\mathrm{sign}\left(r_i^{(\ell +1)}\right)$, $i=1,2,\dots ,N$    $s_j^{(\ell +1)}={\prod }_{i\in \mathcal{Q}\left(c_j\right)}{\widehat{x}}_i^{(\ell +1)}$, $j=1,2,\dots ,M$    $\ell =\ell +1$ end while ${\mathbf{r}}_{out}^{\left(t\right)}=r^{(\ell )}$, ${\mathit{\mu }}_{out}^{\left(t\right)}=\mathit{\mu }$, ${\mathbf{P}}_{out}^{\left(t\right)}=\mathbf{P}$