## 3 Alternant Codes
The parity-check matrix \(H\) of the alternant code is given by the following:  
\[H=\left[\frac{1}{y_{j}}\eta_{j}^{i-1}\right]_{1\leq i\leq r,\ 1\leq j\leq n}.\]  
\(\square\)  
Now, consider the minimum Hamming distance \(d\). With the Singleton bound, for a code of length \(n\), dimension \(k\), and minimum distance \(d\), we have \(d\leq n-k+1\). For an alternant code of length \(n\), dimension \(k\geq n-r\), since it is a subfield subcode of a GRS code over \(F_{q^{m}}\). However, a key result from the theory of alternant codes tells us \(d\geq r+1\). **Justification. ** This is because any codeword \(c=(c_{1},\ldots,c_{n})\in C(n,\eta,y)\subset F_{q^{m}}\) satisfies: \(Hc^{T}=0\). Suppose \(c\) has Hamming weight \(w\leq r\). Then, the set of positions \(j\) such that \(c_{j}\
eq 0\) has size \(\leq r\). The submatrix of \(H\) corresponding to those positions is a \(r\times w\) Vandermonde-like matrix over \(F_{q^{m}}\), and when \(w\leq r\), such a matrix has linearly independent columns, unless \(c_{j}=0\) for all \(j\), contradicting the assumption that \(c\) has weight \(w\). Hence, no non-zero codeword can have weight \(\leq r\), implying \(d\geq r+1\). **Example 1**. : _For the impact of alternant codes constructed over Gaussian integers, let \(p=2\) and \(k=1\), then \(\mathbb{Z}_{2}[i]=\{0,\ 1,\ i,\ 1+i\}\) with \(i^{2}=-1\) being the finite commutative local ring with identity having maximal ideal \(\mathrm{M}=\{0,\ 1+i\}\), then the residue field \(\mathrm{K}=\frac{\mathbb{Z}_{2}[i]}{\mathcal{M}}=\{\overline{0},\overline{1} \}\cong\mathrm{GF}(2)\). If \(g(x)=x^{2}+x+i\), then \(\mu(x^{2}+x+i)=x^{2}+x+1\) it has no rational root and so it is irreducible over \(K\), and \(g(x)=x^{2}+x+i\) is the basic irreducible over \(\mathbb{Z}_{2}[i]\). The given extension ring \(\mathrm{R}=\mathbb{Z}_{2}[i][x]/<x^{2}+x+i>\cong\mathrm{GR}\big{(}2^{4}\big{)}\) is the finite commutative local ring with residue field \(K_{1}\cong\mathrm{GF}\big{(}2^{2}\big{)}\). Then, the maximal cyclic subgroup \(G_{\mathrm{s}}\) has the order \(s=3\). For the purpose of generating the maximal cyclic group \(\mathcal{G}_{3}\), let \(\alpha\) be the root of \(g(x)\) and be of order of \(6\) in set \(\mathbb{R}^{*}\). So, \(\alpha^{2}=\alpha+i\) is the primitive root of \(\mathcal{G}_{3}\). If \(\beta=\alpha+i\), then elements of \(\mathcal{G}_{3}\) are \(\beta=\alpha+i\), \(\beta^{2}=\alpha+i+1\), \(\beta^{3}=1\). Each of the elements of \(\mathcal{G}_{s}\) are all the roots of \(\big{(}x^{3}-1\big{)}\) (modulo \(2\)). Now we choose \(n=3\), \(r=2\), \(\eta=\big{(}1,\ \beta,\ \beta^{2}\big{)}\), and \(y=(1,\ 1,\ 1)\), then \(C(3,\ \eta,\ y)\) is the shortened alternant code with the parity-check matrix \(H\)_\[H= \begin{pmatrix}1&\beta&\beta^{2}\\ 1&\beta^{2}&\beta\end{pmatrix} = \begin{pmatrix}1&\alpha+i&\alpha+(1+i)\\ 1&\alpha+(1+i)&\alpha+i\end{pmatrix} ,\]  
_with a minimum distance \(d\geq 3\). The corresponding parity-check matrix \(H^{\prime}\) of entries from the residue field \(K\cong GF(2)\) is_  
\[H^{\prime}= \begin{pmatrix}1&1&0\\ 0&1&1\\..&..&..\\ 1&0&1\\ 0&1&1\end{pmatrix} .\]  
It is an established fact that the order of every finite field is a power of some prime \(p\). From Corollary 4, \(\mathbb{Z}_{p}[i]\) is a field if and only if \(p\equiv 3(\text{mod }4)\) and \(k=1\). Furthermore, the order of every finite field of Gaussian integers \(\mathbb{Z}_{p}[i]\) with \(p\equiv 3(\text{mod }4)\) will be \(p^{2}\). **Example 2**. : _For the alternant codes construction over Gaussian integers, let \(p=3\) and \(k=1\); then \(\mathbb{Z}_{3}[i]=\{0,\ 1,\ 2,\ i,\ 1+i,\ 2+i,\ 2i,\ 1+2i,\ 2+2i\}\) is a Gaussian field with \(9\) elements, and it is isomorphic to \(\mathcal{GF}(3^{2})\). Let \(f(x)=x^{2}+x+(2+i)\) is a primitive irreducible polynomial over \(\mathbb{Z}_{3}[i]\). Let \(\alpha\) be the root of \(f(x)\); then, \(\alpha\) has order \(80\) in \(\mathbb{Z}_{3}[i]\). The remaining elements of extension field \(\mathbb{Z}_{3}[i]^{2}\) are given in Table 1._  
\[\text{Let}\,\eta=