Optimal Few-SSW Linear Codes and Their Subcode Support Weight Distributions

Few-weight codes have been constructed and studied for many years, since their fascinating relations to finite geometries, strongly regular graphs and Boolean functions. Simplex codes are one-weight <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\left [{{\frac {q^{k}-1}{q-1},k,q^{k-1}}}\right ]_{q}$ </tex-math></inline-formula>-linear codes and they meet all Griesmer bounds on the generalized Hamming weights of linear codes. All the subcodes with dimension r of a <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\left [{{\frac {q^{k}-1}{q-1},k,q^{k-1}}}\right ]_{q}$ </tex-math></inline-formula>-simplex code have the same subcode support weight <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\frac {q^{k-r}(q^{r}-1)}{q-1}$ </tex-math></inline-formula> for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$1\leq r\leq k$ </tex-math></inline-formula>. In this paper, we construct linear codes meeting the Griesmer bound of the r-generalized Hamming weight, such codes do not meet the Griesmer bound of the j-generalized Hamming weight for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$1\leq j\lt r$ </tex-math></inline-formula>. Moreover these codes have only few subcode support weights (few-SSW). The weight distributions and the subcode support weight distributions of these distance-optimal codes are determined. Linear codes constructed in this paper are natural generalizations of distance-optimal few-weight codes.