A Class of Linear Codes and Their Complete Weight Enumerators

Linear codes may have a few weights if their defining sets are chosen properly. Let <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$s $ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$t $ </tex-math></inline-formula> be positive integers. For an odd prime <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p $ </tex-math></inline-formula> and an even integer <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$m $ </tex-math></inline-formula> , let <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q=p^{m} $ </tex-math></inline-formula> , <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$m=2s $ </tex-math></inline-formula> and Tr <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>m</i></sub> (resp. Tr <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>s</i></sub> ) be the absolute trace function from <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbb {F}_{q}$ </tex-math></inline-formula> (resp. <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbb {F}_{p^{s}}$ </tex-math></inline-formula> ) to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbb {F}_{p}$ </tex-math></inline-formula> . In this paper, we define <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$D_{b} =\{ (x_{1},\ldots,x_{t})\in \mathbb {F}_{q}^{t} \backslash \{(0,\ldots,0)\}: \mathrm {Tr}_{m} (x_{1}+\cdots +x_{t})=b\}$ </tex-math></inline-formula> , where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$b \in \mathbb {F}_{p} $ </tex-math></inline-formula> . By employing exponential sums, we demonstrate the complete weight enumerators of a class of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p$ </tex-math></inline-formula> -ary linear codes given by <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$C_{D_{b} }=\{\mathsf {c}(a_{1}, \ldots, a_{t}): a_{1},\ldots,a_{t}\in \mathbb {F}_{p^{s}}\}$ </tex-math></inline-formula> , where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathsf {c}(a_{1}, \ldots, a_{t})=(\mathrm {Tr}_{s}(a_{1}x_{1}^{p^{s}+1}+\cdots +a_{t}x_{t}^{p^{s}+1}))_{(x_{1},\ldots,x_{t})\in D_{b} }$ </tex-math></inline-formula> . Then we get their weight enumerators explicitly, which will give us several linear codes with a few weights. The presented codes are suitable with applications in secret sharing schemes.