On Binary Cyclic Codes with Five Nonzero Weights

Let $q=2^n$, $0\leq k\leq n-1$, $n/\gcd(n,k)$ be odd and $k\neq n/3, 2n/3$. In this paper the value distribution of following exponential sums [\sum\limits_{x\in \bF_q}(-1)^{\mathrm{Tr}1^n(\alpha x^{2^{2k}+1}+\beta x^{2^k+1}+\ga x)}\quad(\alpha,\beta,\ga\in \bF{q})] is determined. As an application, the weight distribution of the binary cyclic code $\cC$, with parity-check polynomial $h_1(x)h_2(x)h_3(x)$ where $h_1(x)$, $h_2(x)$ and $h_3(x)$ are the minimal polynomials of $\pi^{-1}$, $\pi^{-(2^k+1)}$ and $\pi^{-(2^{2k}+1)}$ respectively for a primitive element $\pi$ of $\bF_q$, is also determined.

💡 Research Summary

The paper investigates a family of binary cyclic codes defined by three minimal polynomials and determines their complete weight distribution by evaluating a three‑variable exponential sum. Let $q=2^{n}$, $0\le k\le n-1$, and assume that $s=n/\gcd(n,k)$ is odd while $k\neq n/3,2n/3$. For $\alpha,\beta,\gamma\in\mathbb F_{q}$ define
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