## 4 Decoding Procedure  
In this section, our intention is to demonstrate how the Euclidean algorithm can be used to decode alternant codes. Given \(C(n,\ \eta,\ y)\), an alternant code over \(GF(q)\) with parity-check matrix \(H\), let the code have minimum distance \(d=r+1\). Let \(c=(c_{1},\ c_{2},\ \dots,\ c_{n})\) be the transmitted codeword and \(a=(a_{1},\ a_{2},\ \dots,\ a_{n})\) be the received vector. The error vector is defined as \(e=(e_{1},\ e_{2},\ \dots,\ e_{n})\). Let \(t=\left\lfloor\frac{r}{2}\right\rfloor\) errors that have happened in sites to decode alternant codes  
\[X_{1}=\alpha_{i_{1}},\ X_{2}=\alpha_{i_{2}},\ \dots,\ X_{t}=\alpha_{i_{t}}\]  
In \(\mathcal{G}_{s}\) with error values  
\[Y_{1}=a_{i_{1}},\ Y_{2}=a_{i_{2}},\ \dots,\ Y_{t}=a_{i_{t}}\]  
Firstly, find the syndromes using  
\[S= \begin{pmatrix}y_{1}&y_{2}&\dots&y_{n}\\ y_{1}\alpha_{1}&y_{2}\alpha_{2}&\dots&y_{2}\alpha_{n}\\ &&\ddots\\ y_{1}\alpha_{1}{}^{r-1}&y_{2}\alpha_{2}{}^{r-1}&\dots&y_{n}\alpha_{n}{}^{r-1} \end{pmatrix}\begin{pmatrix}0\\ 0\\ 0\\ \gamma_{t}\\ 0\\ \gamma_{t}\\ 0\end{pmatrix} = \begin{pmatrix}s_{0}\\ s_{1}\\.\\.\\.\\ \gamma_{s-1}\end{pmatrix} ,\]where  
\[S_{\mu}=\sum_{v=1}^{t}\alpha_{i_{v}}^{\mu}a_{i_{v}}y_{i_{v}}=\sum_{v=1}^{t}X_{v}^ {\mu}Y_{v}y_{i_{v}},\text{ For }\mu=0,\ 1,\ldots r-1.\]  
Secondly, find error locator polynomial \(\sigma(z)\) as  
\[\sigma(z)=\prod_{i=1}^{t}(1-X_{i}z)=\sum_{i=0}^{t}\sigma_{i}z^{i},\ \sigma_{0}=1.\]  
Using syndromes, and the error evaluator polynomials  
\[\omega(z)=\sum_{v=1}^{t}Y_{v}w_{i_{v}}\prod_{ \begin{subarray}{c}\mu=1\\ \mu\
eq v\end{subarray} }^{t}(1-X_{\mu}z).\]  
Furthermore,  
\[\frac{\omega(z)}{\sigma(z)}=S(z)(\text{mod }(z^{\prime})),\text{ where }S(z)=\sum_{0}^{r-1}S_{\mu}z^{\mu}.\]  
Using the Euclidean algorithm, the above equation can be solved by  
\[r_{-1}(z)=z^{\prime}\text{ and }r_{0}(z)=S(z).\]  
Let is continue this Euclidean algorithm until \(r_{k}(z)\) such that \(deg(r_{k-1})\leq\frac{r}{2}\) and \(deg(r_{k})\geq\frac{r}{2}-1\). Then, the error locator polynomial and error evaluator polynomial are \(\sigma(z)=\delta U_{k}(z)\) and \(\omega(z)=(-1)^{k}\delta r_{k}(z)\), where constant \(=\delta\). Chosen to make \(\sigma(0)=1\) and \(U_{k}\) satisfies \(r_{k}(z)\equiv(-1)^{k}U_{k}(z)r_{0}(z)\pmod{(r_{-1}(z))}\), and then  
\[\omega(z)\equiv(\sigma(z)S(z))(\text{mod }(z^{\prime})),\ deg(\sigma(z))\leq\frac{1}{2}r, \text{ and }deg(\omega(z))\leq\frac{1}{2}r-1.\]  
Lastly, find the error magnitudes using  
\[\overline{Y}_{\mu}=\frac{\omega\Big{(}X_{\mu}^{-1}\Big{)}}{y_{i_{v}}\prod_{v \
eq\mu}^{t}\left(1-X_{v}X_{\mu}^{-1}\right)}\]  
The pseudo code of the decoding procedure is given in Algorithm 1. ```
Input:
* Alternant code C(n,n,y) over GF(q)
* Received vector a = (a_1, a_2,..., a_n)
* Parity-check matrix H
* Set of support elements \(\alpha\) = (\(\alpha\)_1, \(\alpha\)_2,...,\(\alpha\)_n)
* Set of multipliers y = (y_1, y_2,..., y_n)
* Minimum distance d = r + 1
* Error-correcting capability t = \(\lfloor r/2\rfloor\)
```  
**Algorithm 1**Decoding Alternant Codes Using Euclidean Algorithm  
**Algorithm 2**Decoding Alternant Codes Using Euclidean Algorithm  
**Input:**  
* Alternant code C(n,n,y) over GF(q)
* Received vector a = (a_1, a_2,..., a_n)
* Parity-check matrix H
* Set of support elements \(\alpha\) = (\(\alpha\)_1, \(\alpha\)_2,...,\(\alpha\)_n)
* Set of multipliers y = (y_1, y_2,..., y_n)
* Minimum distance d = r + 1
* Error-correcting capability t = \(\lfloor r/2\rfloor\)  
**Example 4**. : _From Illustration 3.1, the maximal cyclic subgroup \(\mathcal{G}_{3}\), \(\eta=\left(\beta^{2},\,1,\,\beta\right)=\left(\eta_{1},\,\eta_{2},\,\eta_{3}\right)\), \(y=\left(\beta,\,\beta,\,1\right)\) and \(r=2\), so \(t=1\) with length 3 of alternant code \(\mathcal{C}(\eta,\,y)\) over \(\mathbb{Z}_{2}[i]\) with a minimum Hamming distance of at least 3. Let_  
\[H= \begin{pmatrix}\beta&\beta&1\\ \beta^{3}&\beta&\beta\end{pmatrix} ,\]  
_be the parity=check matrix.