# Design and Application of Secret Codes for Learning Medical Data
## 4 Results  
In this section, we present some resultant theory and examples obtained from our constructions. Constructions C and D provide extended codes in the sense of the length and the weight. Moreover, their properties can be shown by combining the properties of the original codes and the construction terminologies, as shown in the following propositions. Proposition 1: _The code \(C_{C}\) in Construction C is a minimal code of length \(2N\) and size \(l\). The weight of each codeword \(y_{i}\) is the sum of the weights of \(x_{1,i}\) and \(x_{2,i}\)._  
Proof. : The length and the size of the code \(C_{C}\) are clear from the definition. The next step is to check the linearity of the new code, that is, whether the equality \(y_{i}+y_{j}\in C_{C}\) for any \(1\leq i,j\leq l\) is satisfied or not. Note that the addition is the vector addition modulo \(2\). Since \(C_{1}\) and \(C_{2}\) are linear codes, it is possible to let \(x_{1,i}+x_{1,j}=x_{1,k}\) for some \(k\) with \(1\leq k\leq l\). Similarly, we have \(x_{2,i}+x_{2,j}=x_{2,k}\). These two equations imply  
\[y_{i}[n]+y_{j}[n]=y_{k}[n]=\left\{ \begin{array}{ll}x_{1,k}\big{[}\big{\lfloor} \frac{n}{2}\big{\rfloor}\big{\rfloor},&\text{if $n$ is even;}\\ x_{2,k}\big{\lfloor}\big{\lfloor}\frac{n}{2}\big{\rfloor}\big{\rfloor},& \text{if $n$ is odd,}\end{array} \right. \tag{8}\]  
by the definition in Construction C. Thus, for all \(1\leq i,j\leq l\), there exists a \(k\) with \(1\leq k\leq l\) such that \(y_{i}+y_{j}=y_{k}\), which implies that \(C_{C}\) is linear. Non-inclusive properties of supports can be also shown using the definition. The support \(\text{supp}(y_{i})\) can be partitioned into \(\text{supp}_{e}(y_{i})\cup\text{supp}_{o}(y_{i})\), where \(\text{supp}_{e}(y_{i})\) (resp. \(\text{supp}_{e}(\mathbf{y}_{i})\)) is the subset of \(\text{supp}(\mathbf{y}_{i})\) with even (resp. odd) indices. It is clear that \(\text{supp}_{e}(\mathbf{y}_{i})\) is equivalent to multiplying \(\text{supp}(x_{1,i})\) by two, and \(\text{supp}_{o}(y_{i})\) is equivalent to adding one after multiplying \(\text{supp}(x_{2,i})\) by two. Thus, \(\text{supp}_{e}(y_{i})\) and \(\text{supp}_{e}(y_{j})\) are not included in one another for any \(1\leq i\
eq j\leq l\). Similarly, \(\text{supp}_{o}(y_{i})\) and \(\text{supp}_{o}(y_{k})\) satisfy the same property. Therefore, it is shown that the code \(C_{C}\) satisfies the conditions of minimal codes. \(\square\)  
Example 1 shows a resultant code applying Construction C to a code present in [20]. Additional parameters of codes with more flexible choices of weights can be obtained by simply changing the index of one code. More parameters from numerical simulations are presented in Section 5. Example 1: _In Example 20 from [20], a minimal code of length \(63\) and size \(128\) is given. The weight distributions for weights \((0,14,30,32,38)\) are given by \((1,1,49,63,14)\). Using two codes from two different primitive elements of the finite field GF(63), we can construct a new code of length \(126\) whose weight distribution is given by \((0,14,30,32,38)\) for weights \((0,28,60,64,76)\) when assuming the order of indices of the codewords are according to the order of weights. If we change the order of combinations, several different codes with different weight distributions are generated. Table 2 compares one example of the code of length \(126\) with the original code of length \(63\)._  
The properties of codes from Construction D can be formulated in the following proposition. Proposition 2: _The code \(C_{D}\) in Construction D is a minimal code of length \(N_{1}N_{2}\) and size \(l_{1}l_{2}\). The weight of each codeword is the product of the weights of two component codewords._ \begin{table}
\begin{tabular}{c c c} \hline \hline
**Constructions** & **Characteristic** & **Linearity** \\ \hline A & Separated supports & may be broken \\ \hline B & Merged supports & may be broken \\ \hline C & Merging two codes & preserved \\ \hline D & Enhanced parameters & preserved \\ \hline \hline \end{tabular}
\end{table}