MinT - Best Known (210−101, 210, s)-Nets in Base 2

Best Known (210−101, 210, s)-Nets in Base 2

(210−101, 210, 56)-Net over F₂ — Constructive and digital

Digital (109, 210, 56)-net over F₂, using

t-expansion [i] based on digital (105, 210, 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]

(210−101, 210, 65)-Net over F₂ — Digital

Digital (109, 210, 65)-net over F₂, using

t-expansion [i] based on digital (95, 210, 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]

(210−101, 210, 232)-Net in Base 2 — Upper bound on s

There is no (109, 210, 233)-net in base 2, because

1 times m-reduction [i] would yield (109, 209, 233)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2²⁰⁹, 233, S₂, 100), but
  - the linear programming bound shows that MÂ â‰¥ 111208 305629 913029 352387 666381 568405 142700 756211 644253 796994 831924 330496 / 125 695773 > 2²⁰⁹ [i]