MinT - Best Known (30, 30+27, s)-Nets in Base 2

Best Known (30, 30+27, s)-Nets in Base 2

(30, 30+27, 24)-Net over F₂ — Constructive and digital

Digital (30, 57, 24)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (8, 21, 12)-net over F₂, using
  - linear code embedding found in a computer search [i]
2. digital (9, 36, 12)-net over F₂, using
  - net from sequence [i] based on digital (9, 11)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 9 and N(F) ≥ 12, using
      - an explicitly constructive algebraic function field [i]

(30, 30+27, 25)-Net over F₂ — Digital

Digital (30, 57, 25)-net over F₂, using

t-expansion [i] based on digital (28, 57, 25)-net over F₂, using
- net from sequence [i] based on digital (28, 24)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 28 and N(F) ≥ 25, using
    - a curve from the manYPoints database [i]

(30, 30+27, 90)-Net in Base 2 — Upper bound on s

There is no (30, 57, 91)-net in base 2, because

1 times m-reduction [i] would yield (30, 56, 91)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2⁵⁶, 91, S₂, 26), but
  - the linear programming bound shows that MÂ â‰¥ 2655 090149 977067 154866 962432 / 34361 762295 > 2⁵⁶ [i]