Construction of a Class of Linear Codes with Four-weight and Six-weight
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摘要: 低重线性码在结合方案、认证码以及秘密共享方案等方面有着极其重要的作用，因而低重线性码的设计一直是线性码的重要研究方向。该文通过选取恰当的定义集，构造了有限域
${F_p}$ (p为奇素数)上的一类四重和六重线性码，利用高斯和确定了码的重量分布，并编写Magma程序进行了验证。结果表明，构造的码中存在关于Singleton界的几乎最佳码。Abstract: Due to the wide applications in association schemes, authentication codes and secret sharing schemes etc., construction of the linear codes with a few weights is an important research topic. A class of linear codes with four-weight and six-weight over finite field${F_p}$ (p is an odd prime) is constructed by a proper selection of the defining set. The explicit weight distribution is obtained using Gauss sums, and some examples from Magma program to illustrate the validity of the conclusions are provided. The results show that these codes include almost optimal codes with respect to Singleton bound.-
Key words:
- Linear codes /
- Authentication codes /
- Weight distribution /
- Gauss sums
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表 1 m为偶数时码CD的重量分布
重量 频数 $0$ $1$ $(p - 1)({p^{m - 2}} + {p^{ - 1}}{G_m})/2$ $p - 1$ $(p - 1)({p^{m - 2}} - {p^{m - 3}})/2$ ${p^{m - 2}} - 1$ $(p - 1)({p^{m - 2}} - {p^{m - 3}} + {p^{ - 2}}{G_m})/2$ $(p - 1)({p^{m - 2}} + {p^{ - 1}}{G_m})$ $(p - 1)({p^{m - 2}} - {p^{m - 3}} + {p^{ - 1}}{G_m})/2$ $(p - 1)({p^{m - 2}} - 1)$ $(p - 1)({p^{m - 2}} - {p^{m - 3}} + {p^{ - 1}}{G_m} + {p^{ - 3}}{G_m}{G^2})/2$ ${A_5}$ $(p - 1)({p^{m - 2}} - {p^{m - 3}} + {p^{ - 1}}{G_m} - {p^{ - 3}}{G_m}{G^2})/2$ ${A_6}$ 
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表 2 m为奇数时码CD的重量分布
重量 频数 $0$ $1$ $\begin{align}{\rm{}}& (p - 1)({p^{m - 2} } - \bar \eta ( - m)\\{\rm{}}& \cdot {p^{ - 2} }{G_m}G)/2\end{align}$ $p - 1$ $(p - 1)({p^{m - 2}} - {p^{m - 3}})/2$ $(p - 1)({p^{m - 2} } - (p - 2)\bar \eta ( - m)\; \\ \cdot{p^{ - 2} }{G_m}G)/2 - 1$ $(p - 1)({p^{m - 2}} - {p^{m - 3}} \\ - \bar \eta ( - m){p^{ - 2}}{G_m}G)/2$ $(p - 1)(2{p^{m - 2} }\; + \bar \eta ( - m)\;\\ \cdot {p^{ - 2} }(p - 2){G_m}G - 1)$ $(p - 1)({p^{m - 2}} - {p^{m - 3}} \\ - 2\bar \eta ( - m){p^{ - 2}}{G_m}G)/2$ $(p - 1)(p - 2)({p^{m - 2} }\; - \bar \eta ( - m)\;\\ \cdot {p^{ - 2} }{G_m}G)/2$
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