MinT - Best Known (33−12, 33, s)-Nets in Base 4

Best Known (33−12, 33, s)-Nets in Base 4

(33−12, 33, 90)-Net over F₄ — Constructive and digital

Digital (21, 33, 90)-net over F₄, using

1 times m-reduction [i] based on digital (21, 34, 90)-net over F₄, using
- trace code for nets [i] based on digital (4, 17, 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]

(33−12, 33, 112)-Net over F₄ — Digital

Digital (21, 33, 112)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4³³, 112, F₄, 12) (dual of [112, 79, 13]-code), using
- 21 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 1, 0, 1, 0, 0, 1, 5 times 0, 1, 8 times 0) [i] based on linear OA(4²⁷, 85, F₄, 12) (dual of [85, 58, 13]-code), using
  - a â€œGraCycâ€ code from Grasslâ€™s database [i]

(33−12, 33, 2039)-Net in Base 4 — Upper bound on s

There is no (21, 33, 2040)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 73 945655 076692 654875 > 4³³ [i]