MinT - Best Known (21, 34, s)-Nets in Base 4

Best Known (21, 34, s)-Nets in Base 4

(21, 34, 90)-Net over F₄ — Constructive and digital

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]

(21, 34, 98)-Net over F₄ — Digital

Digital (21, 34, 98)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4³⁴, 98, F₄, 13) (dual of [98, 64, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(4³⁴, 99, F₄, 13) (dual of [99, 65, 14]-code), using
  - constructionÂ X with VarÅ¡amov bound [i] based on
    1. linear OA(4³², 95, F₄, 13) (dual of [95, 63, 14]-code), using
      - a â€œGraXXâ€ code from Grasslâ€™s database [i]
    2. linear OA(4³², 97, F₄, 12) (dual of [97, 65, 13]-code), using Gilbertâ€“VarÅ¡amov bound and b^mÂ = 4³²Â > V_b^sâˆ’1(kâˆ’1)Â = 16 314816 024161 099533 [i]
    3. linear OA(4⁰, 2, F₄, 0) (dual of [2, 2, 1]-code), using
      - discarding factors / shortening the dual code based on linear OA(4⁰, s, F₄, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(21, 34, 2039)-Net in Base 4 — Upper bound on s

There is no (21, 34, 2040)-net in base 4, because

1 times m-reduction [i] would yield (21, 33, 2040)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 73 945655 076692 654875 > 4³³ [i]