LAUSR

Let F 2 m $\mathbb {F}_{2^{m}}$ be the finite field with 2 m elements, where m is a positive integer. Recently, Heng and Ding in (Finite Fields Appl. 56 :308–331, 2019 ) studied the subfield codes of two families of hyperovel codes and determined the weight distribution of the linear code C a , b = ( ( Tr 1 m ( a f ( x ) + b x ) + c ) x ∈ F 2 m , Tr 1 m ( a ) , Tr 1 m ( b ) ) : a , b ∈ F 2 m , c ∈ F 2 , $$ \mathcal{C}_{a,b}=\left\{((\text{Tr}_{1}^{m}(a f(x)+bx)+c)_{x \in \mathbb{F}_{2^{m}}}, \text{Tr}_{1}^{m}(a), \text{Tr}_{1}^{m}(b)) : a,b \in \mathbb{F}_{2^{m}}, c \in \mathbb{F}_{2}\right\}, $$ for f ( x ) = x 2 and f ( x ) = x 6 with odd m . Let v 2 (⋅) denote the 2-adic order function. This paper investigates more subfield codes of linear codes and obtains the weight distribution of C a , b $\mathcal {C}_{a,b}$ for f ( x ) = x 2 i + 2 j $f(x)=x^{2^{i}+2^{j}}$ , where i , j are nonnegative integers such that v 2 ( m ) ≤ v 2 ( i − j )( i ≥ j ). In addition to this, we further investigate the punctured code of C a , b $\mathcal {C}_{a,b}$ as follows: C a = ( ( Tr 1 m ( a x 2 i + 2 j + b x ) + c ) x ∈ F 2 m , Tr 1 m ( a ) ) : a , b ∈ F 2 m , c ∈ F 2 , $$ \mathcal{C}_{a}=\left\{((\text{Tr}_{1}^{m}(a x^{2^{i}+2^{j}}+bx)+c)_{x \in \mathbb{F}_{2^{m}}}, \text{Tr}_{1}^{m}(a)) : a,b \in \mathbb{F}_{2^{m}}, c \in \mathbb{F}_{2}\right\}, $$ and determine its weight distribution for any nonnegative integers i , j . The parameters of these binary linear codes are new in most cases. Some of the codes and their duals obtained are optimal or almost optimal.