Quasi-perfect linear codes from planar and APN functions

Abstract

Let p be a prime and m be a positive integer with m ? 3. Let f be a mapping from Fpm$\mathbb {F}_{p^{m}}$ to itself and Cf$\mathcal {C}_{f}$ be the linear code of length pm ? 1, whose parity-check matrix has its j-th column ?jf(?j)${\left [\begin {array}{c} \pi ^{j}\\ f(\pi ^{j}) \end {array} \right ]}$, where ? is a primitive element in Fpm$\mathbb {F}_{p^{m}}$ and j = 0, 1, ? , pm ? 2. In the case of p = 2, it is proved that Cf$\mathcal {C}_{f}$ has covering radius 3 when f(x) is a quadratic APN function. This gives a number of binary quasi-perfect codes with minimum distance 5. In the case that p is an odd prime, we show that for all known planar functions f(x), the covering radius of Cf$\mathcal {C}_{f}$ is equal to 2 if m is odd and 3 if m is even. Consequently, several classes of p-ary quasi-perfect codes are derived.