How many weights can a linear code have?

Minjia Shi (1) , Hongwei Zhu (1) , Patrick Sole (2) , Gerard. D. Cohen (3)

1 Anhui University [Hefei]
2 LAGA - Laboratoire Analyse, Géométrie et Applications
3 Télécom ParisTech

We study the combinatorial function L(k, q), the maximum number of nonzero weights a linear code of dimension k over F q can have. We determine it completely for q = 2, and for k = 2, and provide upper and lower bounds in the general case when both k and q are ≥ 3. A refinement L(n, k, q), as well as nonlinear analogues N (M, q) and N (n, M, q), are also introduced and studied.

Domaines: Combinatoire [math.CO]

Minjia Shi, Hongwei Zhu, Patrick Sole, Gerard. D. Cohen. How many weights can a linear code have?. Designs, Codes and Cryptography, 2019, ⟨10.1007/s10623-018-0488-z⟩. ⟨hal-02411615⟩