MinT - Best Known (19, 30, s)-Nets in Base 4

Best Known (19, 30, s)-Nets in Base 4

(19, 30, 90)-Net over F₄ — Constructive and digital

Digital (19, 30, 90)-net over F₄, using

trace code for nets [i] based on digital (4, 15, 45)-net over F₁₆, using
- net from sequence [i] based on digital (4, 44)-sequence over F₁₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 4 and N(F) ≥ 45, using
    - a function field by GarcÃa and Quoos [i]

(19, 30, 115)-Net over F₄ — Digital

Digital (19, 30, 115)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4³⁰, 115, F₄, 11) (dual of [115, 85, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(4³⁰, 120, F₄, 11) (dual of [120, 90, 12]-code), using
  - a â€œBZâ€ code from Brouwerâ€™s database [i]

(19, 30, 2691)-Net in Base 4 — Upper bound on s

There is no (19, 30, 2692)-net in base 4, because

1 times m-reduction [i] would yield (19, 29, 2692)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 288240 062556 611968 > 4²⁹ [i]