MinT - Best Known (40−14, 40, s)-Nets in Base 4

Best Known (40−14, 40, s)-Nets in Base 4

(40−14, 40, 130)-Net over F₄ — Constructive and digital

Digital (26, 40, 130)-net over F₄, using

trace code for nets [i] based on digital (6, 20, 65)-net over F₁₆, using
- net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
    - the Hermitian function field over F₁₆ [i]

(40−14, 40, 152)-Net over F₄ — Digital

Digital (26, 40, 152)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4⁴⁰, 152, F₄, 14) (dual of [152, 112, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(4⁴⁰, 160, F₄, 14) (dual of [160, 120, 15]-code), using
  - a â€œGraâ€ code from Grasslâ€™s database [i]

(40−14, 40, 3100)-Net in Base 4 — Upper bound on s

There is no (26, 40, 3101)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1 211019 336648 413649 050344 > 4⁴⁰ [i]