Optimal Few-GHW 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 Griesmer [qᵏ-1/q-1,k ,qᵏ⁻¹]q-linear codes and they meet all Griesmer bounds of the generalized Hamming weights of linear codes. All the subcodes with dimension r of a [qᵏ-1/q-1,k ,qᵏ⁻¹]q-simplex code have the same subcode support weight qᵏ⁻ʳ(qʳ-1)q-1 for 1≤ r≤ 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≤ j<r. Moreover these codes have only few subcode support weights. The weight distribution 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.