Except for one case over F2, absolutely irreducible quadrics over finite fields with nested rational point sets are identical as varieties, giving precise counts of minimal codewords in projective Reed-Muller codes of order 2.
The parameters of projective Reed–Muller codes.Discrete Math., 81(2):217– 221, 1990
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Maximal quadrics over finite fields and minimal codewords of projective Reed-Muller codes
Except for one case over F2, absolutely irreducible quadrics over finite fields with nested rational point sets are identical as varieties, giving precise counts of minimal codewords in projective Reed-Muller codes of order 2.