Propelinear 1-perfect codes from quadratic functions

Perfect codes obtained by the Vasil’ev–Sch"onheim construction from a linear base code and quadratic switching functions are transitive and, moreover, propelinear. This gives at least $\exp(cN^2)$ propelinear $1$-perfect codes of length $N$ over an arbitrary finite field, while an upper bound on the number of transitive codes is $\exp(C(N\ln N)^2)$. Keywords: perfect code, propelinear code, transitive code, automorphism group, Boolean function.

💡 Research Summary

The paper investigates a rich family of 1‑perfect codes that are simultaneously transitive and propelinear, by exploiting the Vasil’ev–Schönheim (V‑S) construction together with quadratic switching functions. After recalling the basic notions—perfect codes (minimum distance 3, covering radius 1), transitive codes (a single orbit under a subgroup of the automorphism group), and propelinear codes (each codeword determines a permutation combined with a translation, forming a group isomorphic to the code itself)—the authors focus on how to endow the V‑S construction with these strong symmetry properties.

The V‑S construction starts from a linear 1‑perfect code (C_0\subset\mathbb{F}_q^{N_0}) and a function (f:C_0\to\mathbb{F}_q). For each (x\in C_0) the construction inserts the pair ((x,x,f(x))) (and a complementary pair with a flipped last coordinate) into a longer space of length (N=2N_0+1). When (f) is arbitrary, the resulting code (C_f) is still 1‑perfect, but there is no guarantee of any global symmetry. The authors observe that if (f) is a quadratic function, i.e. can be written as \