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# Optimal quinary negacyclic codes with minimum distance four

• * Corresponding author: Yanhai Zhang
This work was supported by Guangxi Natural Science Foundation of China (No. 2018GXNSFBA281019) and the National Natural Science Foundation of China (No. 12061027)
• Based on solutions of certain equations over finite yields, a necessary and sufficient condition for the quinary negacyclic codes with parameters $[\frac{5^m-1}{2},\frac{5^m-1}{2}-2m,4]$ to have generator polynomial $m_{\alpha^3}(x)m_{\alpha^e}(x)$ is provided. Several classes of new optimal quinary negacyclic codes with the same parameters are constructed by analyzing irreducible factors of certain polynomials over finite fields. Moreover, several classes of new optimal quinary negacyclic codes with these parameters and generator polynomial $m_{\alpha}(x)m_{\alpha^e}(x)$ are also presented.

Mathematics Subject Classification: Primary: 94B15, 12E12; Secondary: 11T71.

 Citation: • • Table 1.  Known optimal quinary negacyclic codes $\mathcal{N}_5(1,e)$

 Type e conditions Reference 1) $5^m-2$ $m$ is odd  2) $\frac{5^{k}+1}{2}$ gcd$(k,2m)=1$  3) $\frac{2\cdot 5^m-1}{3}$ $m$ is odd  4) $5^k+2$ $m=2k$, $k$ is even  5) $\frac{5^{\frac{m+1}{2}}-1}{2}+\frac{5^m-1}{4}$ $m>1$ is odd  6) $\frac{5^m-1}{2}-3$ $m>1$ is odd 

Table 2.  Optimal quinary negacyclic codes $\mathcal{N}_5(1,u)$

 Type u conditions Reference ⅰ) $\frac{5^m-1}{2}+\frac{5^t+1}{2}$ $1\leq t Table 3. Optimal quinary negacyclic codes$ \mathcal{N}_5(3,u) $ Type u conditions Reference ⅰ)$ 5^m-4  m > 1\; {\rm{is\;odd}} $Section 4.2 ⅱ)$ \frac{5^m-1}{2}-1  m > 1\; {\rm{is\;odd}} $Section 4.3 ⅲ)$ \frac{5^m-1}{2}-9  m > 1\; {\rm{is\;odd}} $Section 4.3 Table 4. Weight distributions of$ \mathcal{N}_5(3,121)^{\bot}, \mathcal{N}_5(3,61)^{\bot} \;and\;\mathcal{N}_5(3,53)^{\bot} $$ \mathcal{N}_5(3,121)^{\bot}  \mathcal{N}_5(3,61)^{\bot}  \mathcal{N}_5(3,53)^{\bot} \$ Weight Frequency Weight Frequency Weight Frequency 0 1 0 1 0 1 43 744 44 1488 44 1488 46 2232 46 2232 46 2232 47 744 48 744 48 744 49 2976 49 2232 49 2232 50 496 50 3224 50 3224 51 744 51 1488 51 1488 52 2976 52 1736 52 1736 53 744 53 1736 53 1736 54 744 57 744 57 744 48 2232 55 248 56 744
• Tables(4)

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