## 5 Comparative Analysis and Discussion  
This section provides a comparative study of alternant codes over Galois rings and Gaussian rings, referring to the concept from [PERSON] [4] and [PERSON] [8]. Two-dimensional vector algebra for local rings and fields is divided into two cases: Gaussian and Eisenstein. Eisenstein is discussed in our previously published article [16] in 2025. We cannot compare the proposed work with Eisenstein integers [16], because both Gaussian and Eisenstein integers have different congruence classes dependent on parameters \(p\) and \(k\). As a type of error-correcting code, alternant codes are studied in terms of various parameters, including the code length \(n\), dimension \(k^{\prime}\), minimum distance \(d\), ability to detect errors \((d-1)\), ability to correct errors \(\left(t=floor\left(\frac{d-1}{2}\right)\right)\), code rate \(R=k^{\prime}/n\) and the number of codewords \(|C|\). A comparative study of alternant codes over Gaussian rings \(\mathbb{Z}_{p}[i]^{s}\) and Galois rings \(GR\left(p^{k},\ s\right)\) is given in Table 2.  
Based on the tabulated results, we can see that alternant codes over Gaussian rings provide a much larger codebook size \(|\ C\ |\) compared to their counterpart over Galois rings, with the same value of code parameters n, k, and d. For instance:  
* When \(n=3,\ k^{\prime}=1,\ d=3\), the minimum distance and the code rate \((R=0.3333)\) are the same when using \(\mathbb{Z}_{2}[i]^{2}\) and \(GR\left(2^{2},\ 2\right)\); however, the code over the Gaussian ring has four times as many codewords (\(|C|=16\) versus \(|C|=4\)). * In the case of \(n=9,\ k^{\prime}=6,\ d\geq 4\), \(\mathbb{Z}_{3}[i]^{2}\) and \(GR\left(3^{2},\ 2\right)\) have the same structural parameters and error correction ability, but the Gaussian ring construction has twice the number of codewords \(\left(3^{24}\ \text{vs.}\ 3^{12}\right)\). * A greater advantage is seen in larger lengths like \(n=80,\ k^{\prime}=76\), where both \(\mathbb{Z}_{3^{2}}[i]^{2}\) and \(GR\left(3^{2},\ 4\right)\) provide five or more minimum distances and approximately the same code rate \((R=0.95)\) but the code over the Gaussian ring has exponential growth in the number of codewords, with \(|C|=3^{304}\), whereas the Galois ring has \(\left|C\right|=3^{152}\). A direct consequence of this doubling of the codeword length in all examples is that this suggests a more abundant codebook and, consequently, greater redundancy and more error resilience with Gaussian rings. The algebraic view as a Gaussian integer, especially the complex units and extra symmetry, enables more valid codewords with the same constraints. There is, however, a trade-off with this benefit: a bigger codebook means increased complexity in decoding and possibly lower transmission speeds because of the symbol space. Nevertheless, where error tolerance and robustness are valued, as in deep-space communication, military systems, or quantum-safe protocols, it is very desirable to use alternant codes instead of Gaussian rings. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Structure** & **Reference** & \(n\) & \(\mathbf{k}^{\prime}\) & \(d\) & \(d\mathbf{-1}\) & \(t\) & \(R\) & \(1\,C1\) \\ \hline \(\mathbb{Z}_{2}[i]^{2}\) & Proposed & 3 & 1 & 3 & 2 & 1 & 0.3333 & 16 \\ \hline \(\mathbb{Z}_{3}[i]^{2}\) & Proposed & 9 & 6 & \(\geq\)4 & \(\geq\)3 & \(\geq\)1 & 0.6667 & \(3^{24}\) \\ \hline \(\mathbb{Z}_{3^{2}}[i]^{2}\) & Proposed & 80 & 76 & \(\geq\)5 & \(\geq\)4 & \(\geq\)2 & 0.95 & \(3^{304}\) \\ \hline \(GR\left(2^{2},\ 2\right)\) & [11, 22] & 3 & 1 & 3 & 2 & 1 & 0.3333 & 4 \\ \hline \(GR\left(3^{2},\ 2\right)\) & [11, 22] & 9 & 6 & \(\geq\)4 & \(\geq\)3 & \(\geq\)1 & 0.6667 & \(3^{12}\) \\ \hline \(GR\left(3^{2},\ 4\right)\) & [11, 22] & 80 & 76 & \(\geq\)5 & \(\geq\)4 & \(\geq\)2 & 0.95 & \(3^{152}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparative analysis of alternant codes over Galois rings and Gaussian rings.