Optimal Few-SSW Linear Codes and Their Subcode Support Weight Distributions
Xu Pan, Hao Chen, Hongwei Liu, Shengwei Liu · IEEE Transactions on Information Theory · 2024
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$\left [{{\frac {q^{k}-1}{q-1},k,q^{k-1}}}\right ]_{q}$-linear codes and they meet all Griesmer bounds on the generalized Hamming weights of linear codes. All the subcodes with dimension r of a$\left [{{\frac {q^{k}-1}{q-1},k,q^{k-1}}}\right ]_{q}$-simplex code have the same subcode support weight$\frac {q^{k-r}(q^{r}-1)}{q-1}$for$1\leq r\leq k$. 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$1\leq j\lt r$. 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.