## 3 Alternant Codes  
We explain the construction technique of alternant codes over the finite commutative local rings of Gaussian integers with identity. Initially, we list some data from the basic theory of commutative rings. Thus, \(\mathbb{Z}[i]_{n}\) is assumed to be a finite local commutative ring with unity, \(M\) is the maximal ideal of rings \(\mathbb{Z}[i]_{n}\), and the residue field is \(\mathrm{K}=\frac{\mathbb{Z}[i]_{n}}{\mathrm{M}}\cong GF(q)\), where \(q=p^{m}\ m\in\mathbb{Z}^{+}\) and \(p\) is a prime. Let \(\mathbb{Z}[i]_{n}[\mathrm{x}]=\left\{\sum a_{j}x^{j}:\ \forall a_{j}\in \mathbb{Z}[i]_{n}\right\}\). Let \(g(x)\) be a monic polynomial of degree \(s^{\prime}\) in \(\mathbb{Z}[i]_{n}[\mathrm{x}]\), such that \(\mu(g(x))\) is irreducible in \(\mathrm{K}[x]\), where \(\mu\) is a natural projection. Moreover, we would also conclude that \(g(x)\) is irreducible because it gave a factorization of \(h_{j}(x)\) in terms of lower-order cyclotomic polynomials and is irreducible in \(\mathbb{Z}[i]_{n}[\mathrm{x}]\). Suppose \(R=\frac{\mathbb{Z}[i]_{n}[\mathrm{x}]}{<g(x)>}\cong GR(p^{k},\ s^{\prime})\) is a finite commutative local ring with unity and is the Gaussian extension of \(\mathbb{Z}[i]_{n}\) of degree \(s^{\prime}\). Its residue field will be \(K_{1}=\frac{R}{\mathrm{M}_{1}}\cong GF\Big{(}p^{ms^{\prime}}\Big{)}\), where \(\overline{\mathrm{M}_{1}}\) is the maximal ideal of \(R\). Let \(K_{1}\)* be the unit element \(K_{1}\), and its order is \(p^{ms^{\prime}}-1\). Suppose \(R^{*}\) is the multiplicative group of \(R\); then, as is customary with all multiplicative groups, it must be a direct product of cyclic groups. The cyclic group is the maximal cyclic subgroup of the group of units in \(R\). The elements of that maximal cyclic group \(G_{s}\) are the roots of the polynomial \(x^{s}-1\) for some \(s\), where \(gcd(s,\ p)=1\), it is isomorphic to \(K_{1}^{*}\). The cyclic group that is maximal then is denoted \(G_{s}\) and has the order \(s=p^{ms^{\prime}}-1\). It has long been known that a parity=check matrix \(H\) with \(r\) rows and \(n\) columns of elements from \(G_{s}\) of \(GF(q^{m})\) can uniquely characterize an \((n,\ k_{0})\) linear code over \(GF(q)\), where \(m,\ r\in\mathbb{Z}^{+}\) and \(rm\geq n-k_{0}\). Definition 1: Given maximal cyclic subgroup \(G_{s}\) of \(\mathbb{Z}_{p^{k}}[i]^{m}\), assume that the locator polynomial \(\eta=(\alpha_{1},\ \alpha_{2},\ \ldots,\ \alpha_{n})\) consists of distinct elements of \(G_{s}\) and \(y=(y_{1},\ y_{2},\ldots,\ y_{n})\) is an arbitrary vector consisting of the elements of \(G_{s}\). Then, a parity-check matrix \(H\) of shortened alternant code \(C(n,\ \eta,\ y)\) of length \(n\leq s\) over \(\mathbb{Z}_{p^{k}}[i]\) is defined as follows:\[H=\left( \begin{array}{ccccc}y_{1}&y_{2}&\ldots&y_{n}\\ y_{1}\alpha_{1}&y_{2}\alpha_{2}&\ldots&y_{2}\alpha_{n}\\ &&&\ddots\\ y_{1}{\alpha_{1}}^{r-1}&y_{2}{\alpha_{2}}^{r-1}&\ldots&y_{n}{\alpha_{n}}^{r-1} \end{array} \right),\]  
_where \(r\in\mathbb{Z}^{+}\) and \(r=n-k_{0}\). Additionally, \(C(n,\ \eta,\ y)\) is a linear alternant code \([n,\ k_{0},\ d]\) over \(K=GF(p^{m})\), with \(n-mr\leq k_{0}\leq n-r\). The number of parity-check symbols is at most \(mr\). Apart from it, it is possible to estimate the minimum Hamming distance for \(C(n,\ \eta,\ y)\) from the parity-check matrix._  
**Theorem 6**. : _It can also be seen that the minimum Hamming distance of alternant code \(C(n,\eta,y)\) will be at least \(d\geq r+1\)._  
Proof. : Let \(C(n,\eta,y)\) be an alternant code of length \(n\), where \(\eta=(\eta_{1},\ldots,\eta_{n})\in F_{q^{m}}\) is a vector of distinct elements, \(y=(y_{1},\ldots,y_{n})\in F_{q^{m}}\) is a non-zero vector, and \(r\) is the designed error-correcting capability, i.e., the alternant code is derived from a Generalized Reed-Solomon (GRS) code of dimension \(n-r\).