## 2 Gaussian Integers  
By following Section 2 in [20], Gaussian integers (GI) ring \(\mathbb{Z}[i]=\{b_{0}+b_{1}i:\ b_{0},\ b_{1}\in\mathbb{Z}\}\) is a subset of complex numbers, with multiplicative unit 1 and \(i^{2}=-1\). Let \(g=a+bi\) be a Gaussian integer with conjugate \(\overline{g}=a-bi\), and then the norm of Gaussian integer is \(N(g)\) which is the product of a Gaussian integer with its norm (sum of squares of real part and coefficient of imaginary part) as \(g\overline{g}=a^{2}+b^{2}\). For example, \(g=2+3i\) is a Gaussian integer; its conjugate is \(\overline{g}=2-3i\). The norm of \(g\) is \(N(g)=(2+3i)(2-3i)=2^{2}+3^{2}=13\). Multiplication of the norm of Gaussian integers satisfies the \(N(g_{1}.g_{2}\ldots g_{n})=N(g_{1}).N(g_{2})\ldots N(g_{n})\), where \(g_{1},\ g_{2},\ \ldots,\ g_{n}\in\mathbb{Z}[i]\). **Theorem 1** ([21], Theorem 2.3.2.). : _The Gaussian ring \(\mathbb{Z}[i]\) is an integral domain._  
**Corollary 1** ([21], Corollary 2.4.2.1.). : _The integral domain \(\mathbb{Z}[i]\) is an Euclidean domain._  
**Lemma 1** ([21], Lemma 3.1.1.). : _If \(\mathbb{Z}[i]\) is a Euclidean domain, then \(\mathbb{Z}[i]\) is a principal ideal domain._  
**Corollary 2** ([21], Corollary 6.2.4.1.). : _Let \(p\in\mathbb{Z}\) be prime, with \(p\equiv 1(\mathrm{mod}\ 4)\); then \(p\) can be factored into a product of the Gaussian prime in \(\mathbb{Z}[i]\)._  
**Corollary 3** ([21], Corollary 6.2.4.2.). : _Let \(p\in\mathbb{Z}\) be prime, with \(p\equiv 1(\mathrm{mod}\ 4)\); then \(p\) is not prime in \(\mathbb{Z}[i]\)._  
**Theorem 2** ([21], Theorem 6.2.5.). : _Let \(p\in\mathbb{Z}\) be prime, with \(p\equiv 3(\mathrm{mod}\ 4)\); then \(p\) is irreducible in \(\mathbb{Z}[i]\)._  
**Lemma 2** (Section 2 in [22]). : _Let \(g=a+ib\in\mathbb{Z}_{n}[i]\) is unit if and only if \(N(g)\) is a unit in \(\mathbb{Z}_{n}\)._  
We observe that \(\{\pm 1,\ \pm i\}\) are units in \(\mathbb{Z}[i]\) and that Gaussian primes are the usual integers \(p\) such that \(p\equiv 3(\mathrm{mod}\ 4)\) and those Gaussian integers \(\mathbb{Z}[i]\) such that \(N(z)=p\). It can be seen easily that the ring \(\mathbb{Z}[i]/n\mathbb{Z}[i]\) is canonically isomorphic to \(\mathbb{Z}[i]_{n}=\{a+ib:a,\ b\in\mathbb{Z}_{n}\}\). This ring is denoted by \(\mathbb{Z}[i]_{n}\), and it is a principal ideal ring, Definition 3.7. [22]; it suffices to fix \(n\in\mathbb{Z}\) rather than fix \(n\), such that \(a+ib\). First, when \(n\) is of the form \(p^{k}\), where \(p\) is a prime integer and \(k>0\). Since our main purpose is fields and local commutative rings of Gaussian integers, in this case, it may be noted that the ring \(\mathbb{Z}_{p^{k}}[i]\) may not always be a local ring. Likewise, \(\mathbb{Z}_{p}[i]\) is not a field for every prime \(p\). We begin with the description of \(\mathbb{Z}_{p^{k}}[i]\) for primes \(p\) such that \(\mathbb{Z}_{p^{k}}[i]\) is a local ring or field. **Theorem 3** ([23]). : _The ring \(\mathbb{Z}_{2^{k}}[i]\) is local for every positive integer \(k\)._  
**Theorem 4** ([23]). : _The ring \(\mathbb{Z}_{p^{k}}[i]\) is local for every positive integer \(k\) and \(p\equiv 3(\mathrm{mod}\ 4)\)._  
**Corollary 4** ([6]). : _Let \(\mathbb{Z}_{p^{k}}[i]\) be a Gaussian field if \(p\equiv 3(\mathrm{mod}\ 4)\) and \(k=1\)._The following results characterize the unit elements of \(\mathbb{Z}[i]_{n}=\mathbb{Z}_{p^{k}}[i]\). From Section 2 in [22];  
(i) \(U\Big{(}\mathbb{Z}_{p^{k}}[i]\Big{)}=2^{2k-1}\), if \(p\equiv 2\). (ii) \(U\Big{(}\mathbb{Z}_{p^{k}}[i]\Big{)}=\Big{(}p^{k}-p^{k-1}\Big{)}^{2}\), if \(p\equiv 1(\text{mod }4)\). (iii) \(U\Big{(}\mathbb{Z}_{p^{k}}[i]\Big{)}=p^{2k}-p^{2k-2}\), if \(p\equiv 3(\text{mod }4)\). Impact of Alternant Codes in Modern Technology: Control and efficient data transfer in a constantly changing environment are crucial for the effectiveness of a number of exciting modern technological solutions, such as the 5G cellular network, computing in the cloud, automobiles without drivers, and the Internet of Things (IoT).