MinT - Best Known (232−128, 232, s)-Nets in Base 2

Best Known (232−128, 232, s)-Nets in Base 2

(232−128, 232, 55)-Net over F₂ — Constructive and digital

Digital (104, 232, 55)-net over F₂, using

t-expansion [i] based on digital (100, 232, 55)-net over F₂, using
- net from sequence [i] based on digital (100, 54)-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 6 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]

(232−128, 232, 65)-Net over F₂ — Digital

Digital (104, 232, 65)-net over F₂, using

t-expansion [i] based on digital (95, 232, 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]

(232−128, 232, 218)-Net in Base 2 — Upper bound on s

There is no (104, 232, 219)-net in base 2, because

26 times m-reduction [i] would yield (104, 206, 219)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2²⁰⁶, 219, S₂, 102), but
  - the linear programming bound shows that MÂ â‰¥ 78984 218751 417890 023438 520762 752824 239171 321550 411833 440733 233152 / 767 > 2²⁰⁶ [i]