A note on small weight codewords of projective geometric codes and on the smallest sets of even type

02/09/2023

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by   Sam Adriaensen, et al.

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In this paper, we study the codes 𝒞_k(n,q) arising from the incidence of points and k-spaces in PG(n,q) over the field 𝔽_p, with q = p^h, p prime. We classify all codewords of minimum weight of the dual code 𝒞_k(n,q)^⊥ in case q ∈{4,8}. This is equivalent to classifying the smallest sets of even type in PG(n,q) for q ∈{4,8}. We also provide shorter proofs for some already known results, namely of the best known lower bound on the minimum weight of 𝒞_k(n,q)^⊥ for general values of q, and of the classification of all codewords of 𝒞_n-1(n,q) of weight up to 2q^n-1.