Towards the Complete Determination of Next-to-Minimal Weights of Projective Reed-Muller Codes

Relative generalized Hamming weights of evaluation codes

“…Cherdieu, Rolland, Geil, Bruen, and Leduc studied the weight distribution of generalized Reed-Muller codes in [4,11,15,29,37]. More recently, Carvalho and Neumann have determined the next-to-minimal weights of affine Cartesian codes, binary projective Reed-Muller codes, and projective Reed-Muller codes [5,6,7,8,9,10]. In this work, we study the next-to-minimal weights of certain evaluation codes called toric codes over hypersimplices.…”