MinT - Best Known (66, 66+63, s)-Nets in Base 2

Best Known (66, 66+63, s)-Nets in Base 2

(66, 66+63, 43)-Net over F₂ — Constructive and digital

Digital (66, 129, 43)-net over F₂, using

t-expansion [i] based on digital (59, 129, 43)-net over F₂, using
- net from sequence [i] based on digital (59, 42)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 54, N(F)Â =Â 42, and 1 place with degree 6 [i] based on function field F/F₂ with g(F) = 54 and N(F) ≥ 42, using an explicitly constructive algebraic function field [i]

(66, 66+63, 48)-Net over F₂ — Digital

Digital (66, 129, 48)-net over F₂, using

t-expansion [i] based on digital (65, 129, 48)-net over F₂, using
- net from sequence [i] based on digital (65, 47)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 65 and N(F) ≥ 48, using
    - Drinfeld modules of rank 1 [i]

(66, 66+63, 145)-Net in Base 2 — Upper bound on s

There is no (66, 129, 146)-net in base 2, because

1 times m-reduction [i] would yield (66, 128, 146)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2¹²⁸, 146, S₂, 62), but
  - the linear programming bound shows that MÂ â‰¥ 10 546031 115613 724859 656905 833525 360409 444352 / 30229 > 2¹²⁸ [i]