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# The weight distribution of a class of p-ary cyclic codes and their applications

• * Corresponding author: Shixin Zhu
This research is supported in part by the National Natural Science Foundation of China under Project 61572168, Project 61772168 and Project 11501156, the Natural Science Foundation of Anhui Province under Grant 1808085MA15 and the Key University Science Research Project of Anhui Province under Grant KJ2018A0497.
• Cyclic codes over finite field have been studied for decades due to their wide applications in communication and storage systems. However their weight distributions are known only in a few cases. In this paper, we investigate a class of $p$-ary cyclic codes whose duals have three zeros, where $p$ is an odd prime. The weight distributions of the class of cyclic codes for all distinct cases are determined explicitly. The results indicate that these codes contain five-weight codes, seven-weight codes and eleven-weight codes. Some of these codes are optimal. Moreover, the covering structures of the class of codes are considered and being used to construct secret sharing schemes.

Mathematics Subject Classification: Primary: 94B05, 11T71; Secondary: 94B15.

 Citation: • • Table Ⅰ.  Weight distribution of the cyclic code C for odd m in Theorem 3.1

 Hamming Weight $i$ Frequency $A_i$ $0$ $1$ $(p-1)p^{m-1}$ $(p^m-1)(p^{m-1}+1)$ $(p-1)p^{m-1}-1$ $(p^m-1)(p^{m-1}+1)(p-1)$ $(p-1)p^{m-1}-p^{\frac{m-1}{2}}$ $\frac{1}{2}(p^m-1)(p^{m-1}+p^{\frac{m-1}{2}})(p-1)$ $(p-1)p^{m-1}-p^{\frac{m-1}{2}}-1$ $\frac{1}{2}(p^m-1)\big((p-1)p^{m-1}-p^{\frac{m-1}{2}}\big)(p-1)$ $(p-1)p^{m-1}+p^{\frac{m-1}{2}}$ $\frac{1}{2}(p^m-1)(p^{m-1}-p^{\frac{m-1}{2}})(p-1)$ $(p-1)p^{m-1}+p^{\frac{m-1}{2}}-1$ $\frac{1}{2}(p^m-1)\big((p-1)p^{m-1}+p^{\frac{m-1}{2}}\big)(p-1)$ $p^{m}-1$ $p-1$

Table Ⅱ.  Weight distribution of the cyclic code C for even m in Theorem 3.1

 Hamming Weight $i$ Frequency $A_i$ $0$ $1$ $(p-1)p^{m-1}$ $p^m-1$ $(p-1)p^{m-1}-1$ $(p^m-1)(p-1)$ $(p-1)(p^{m-1}-p^{\frac{m-2}{2}})$ $\frac{1}{2}(p^m-1)(p^{m-1}+(p-1)p^{\frac{m-2}{2}})$ $(p-1)(p^{m-1}-p^{\frac{m-2}{2}})-1$ $\frac{1}{2}(p^m-1)(p^{m-1}-p^{\frac{m-2}{2}})(p-1)$ $(p-1)p^{m-1}-p^{\frac{m-2}{2}}$ $\frac{1}{2}(p^m-1)(p^{m-1}+p^{\frac{m-2}{2}})(p-1)$ $(p-1)p^{m-1}-p^{\frac{m-2}{2}}-1$ $\frac{1}{2}(p^m-1)((p-1)p^{m-1}-p^{\frac{m-2}{2}})(p-1)$ $(p-1)p^{m-1}+p^{\frac{m-2}{2}}$ $\frac{1}{2}(p^m-1)(p^{m-1}-p^{\frac{m-2}{2}})(p-1)$ $(p-1)p^{m-1}+p^{\frac{m-2}{2}}-1$ $\frac{1}{2}(p^m-1)((p-1)p^{m-1}+p^{\frac{m-2}{2}})(p-1)$ $(p-1)(p^{m-1}+p^{\frac{m-2}{2}})$ $\frac{1}{2}(p^m-1)(p^{m-1}+(p-1)p^{\frac{m-2}{2}})$ $(p-1)(p^{m-1}+p^{\frac{m-2}{2}})-1$ $\frac{1}{2}(p^m-1)(p^{m-1}+p^{\frac{m-2}{2}})(p-1)$ $p^{m}-1$ $p-1$

Table Ⅲ.  Weight distribution of the cyclic code C for even m in Theorem 3.2

 Hamming Weight $i$ Frequency $A_i$ $0$ $1$ $(p-1)p^{m-1}$ $(p^m-1)(1+p^{m-d}-p^{m-2d})$ $(p-1)p^{m-1}-1$ $(p-1)(p^m-1)(1+p^{m-d}-p^{m-2d})$ $(p-1)(p^{m-1}-p^{\frac{m+2d-2}{2}})$ $(p^{m-2d-1}-(p-1)p^{\frac{m-2d-2}{2}})\dfrac{p^m-1}{p^d+1}$ $(p-1)(p^{m-1}-p^{\frac{m+2d-2}{2}})-1$ $(p-1)(p^{m-2d-1}-p^{\frac{m-2d-2}{2}})\dfrac{p^m-1}{p^d+1}$ $(p-1)p^{m-1}-p^{\frac{m-2}{2}}$ $(p-1)(p^{m-1}+p^{\frac{m-2}{2}})\dfrac{p^d(p^m-1)}{p^d+1}$ $(p-1)p^{m-1}-p^{\frac{m-2}{2}}-1$ $(p-1)((p-1)p^{m-1}-p^{\frac{m-2}{2}})\dfrac{p^d(p^m-1)}{p^d+1}$ $(p-1)p^{m-1}+p^{\frac{m+2d-2}{2}}$ $(p-1)(p^{m-2d-1}-p^{\frac{m-2d-2}{2}})\dfrac{p^m-1}{p^d+1}$ $(p-1)p^{m-1}+p^{\frac{m+2d-2}{2}}-1$ $(p-1)((p-1)p^{m-2d-1}+ p^{\frac{m-2d-2}{2}})\dfrac{p^m-1}{p^d+1}$ $(p-1)(p^{m-1}+p^{\frac{m-2}{2}})$ $(p^{m-1}-(p-1)p^{\frac{m-2}{2}})\dfrac{p^d(p^m-1)}{p^d+1}$ $(p-1)(p^{m-1}+p^{\frac{m-2}{2}})-1$ $(p-1)(p^{m-1}+p^{\frac{m-2}{2}})\dfrac{p^d(p^m-1)}{p^d+1}$ $p^{m}-1$ $p-1$

Table Ⅳ.  Weight distribution of the cyclic code C for even m in Theorem 3.2

 Hamming Weight $i$ Frequency $A_i$ $0$ $1$ $(p-1)p^{m-1}$ $(p^m-1)(1+p^{m-d}-p^{m-2d})$ $(p-1)p^{m-1}-1$ $(p-1)(p^m-1)(1+p^{m-d}-p^{m-2d})$ $(p-1)(p^{m-1}-p^{\frac{m-2}{2}})$ $(p^{m-1}+(p-1)p^{\frac{m-2}{2}})\dfrac{p^d(p^m-1)}{p^d+1}$ $(p-1)(p^{m-1}-p^{\frac{m-2}{2}})-1$ $(p-1)(p^{m-1}-p^{\frac{m-2}{2}})\dfrac{p^d(p^m-1)}{p^d+1}$ $(p-1)p^{m-1}-p^{\frac{m+2d-2}{2}}$ $(p-1)(p^{m-2d-1}+p^{\frac{m-2d-2}{2}})\dfrac{p^m-1}{p^d+1}$ $(p-1)p^{m-1}-p^{\frac{m+2d-2}{2}}-1$ $(p-1)((p-1)p^{m-2d-1}-p^{\frac{m-2d-2}{2}})\dfrac{(p^m-1)}{p^d+1}$ $(p-1)p^{m-1}+p^{\frac{m-2}{2}}$ $(p-1)(p^{m-1}-p^{\frac{m-2}{2}})\dfrac{p^d(p^m-1)}{p^d+1}$ $(p-1)p^{m-1}+p^{\frac{m-2}{2}}-1$ $(p-1)((p-1)p^{m-1}+p^{\frac{m-2}{2}})\dfrac{p^d(p^m-1)}{p^d+1}$ $(p-1)(p^{m-1}+p^{\frac{m+2d-2}{2}})$ $(p^{m-2d-1}-(p-1)p^{\frac{m-2d-2}{2}})\dfrac{p^m-1}{p^d+1}$ $(p-1)(p^{m-1}+p^{\frac{m+2d-2}{2}})-1$ $(p-1)(p^{m-2d-1}+p^{\frac{m-2d-2}{2}})\dfrac{p^m-1}{p^d+1}$ $p^{m}-1$ $p-1$

Table Ⅴ.  Weight distribution of the cyclic code C for even m in Corollary 1

 Hamming Weight $i$ Frequency $A_i$ $0$ $1$ $(p-1)p^{m-1}$ $(p^m-1)$ $(p-1)p^{m-1}-1$ $(p-1)(p^m-1)$ $(p-1)p^{m-1}-p^{\frac{m-2}{2}}$ $(p-1)(p^{m-1}+p^{\frac{m-2}{2}})(p^d-1)$ $(p-1)p^{m-1}-p^{\frac{m-2}{2}}-1$ $(p-1)((p-1)p^{m-1}-p^{\frac{m-2}{2}})(p^d-1)$ $(p-1)(p^{m-1}+p^{\frac{m-2}{2}})$ $(p^{m-1}-(p-1)p^{\frac{m-2}{2}})(p^d-1)$ $(p-1)(p^{m-1}+p^{\frac{m-2}{2}})-1$ $(p-1)(p^{m-1}+p^{\frac{m-2}{2}})(p^d-1)$ $p^{m}-1$ $p-1$

Table Ⅵ.  Weight distribution of the cyclic code C for even m in Corollary 2

 Hamming Weight $i$ Frequency $A_i$ $0$ $1$ $(p-1)p$ $(p^2-1)$ $p(p-1)-1$ $2(p-1)(p^2-1)$ $(p-1)p-2$ $(p-1)^2(p^2-p-1)$ $p^2-2$ $(p-1)(p^2-1)$ $p^2-1$ $2(p-1)$
• Tables(6)

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