MinT - Best Known (212−104, 212, s)-Nets in Base 2

Best Known (212−104, 212, s)-Nets in Base 2

(212−104, 212, 56)-Net over F₂ — Constructive and digital

Digital (108, 212, 56)-net over F₂, using

t-expansion [i] based on digital (105, 212, 56)-net over F₂, using
- net from sequence [i] based on digital (105, 55)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 69, N(F)Â =Â 48, 1 place with degree 2, and 7 places with degree 6 [i] based on function field F/F₂ with g(F) = 69 and N(F) ≥ 48, using an explicitly constructive algebraic function field [i]

(212−104, 212, 65)-Net over F₂ — Digital

Digital (108, 212, 65)-net over F₂, using

t-expansion [i] based on digital (95, 212, 65)-net over F₂, using
- net from sequence [i] based on digital (95, 64)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 95 and N(F) ≥ 65, using
    - Drinfeld modules of rank 1 [i]

(212−104, 212, 228)-Net in Base 2 — Upper bound on s

There is no (108, 212, 229)-net in base 2, because

2 times m-reduction [i] would yield (108, 210, 229)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2²¹⁰, 229, S₂, 102), but
  - the linear programming bound shows that MÂ â‰¥ 6611 926920 118694 409642 085450 091564 422709 509669 628075 400990 660413 620224 / 3 180749 > 2²¹⁰ [i]