MinT - Best Known (219−101, 219, s)-Nets in Base 2

Best Known (219−101, 219, s)-Nets in Base 2

(219−101, 219, 57)-Net over F₂ — Constructive and digital

Digital (118, 219, 57)-net over F₂, using

t-expansion [i] based on digital (110, 219, 57)-net over F₂, using
- net from sequence [i] based on digital (110, 56)-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 8 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]

(219−101, 219, 73)-Net over F₂ — Digital

Digital (118, 219, 73)-net over F₂, using

t-expansion [i] based on digital (114, 219, 73)-net over F₂, using
- net from sequence [i] based on digital (114, 72)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 114 and N(F) ≥ 73, using
    - Drinfeld modules of rank 1 [i]

(219−101, 219, 259)-Net in Base 2 — Upper bound on s

There is no (118, 219, 260)-net in base 2, because

1 times m-reduction [i] would yield (118, 218, 260)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2²¹⁸, 260, S₂, 100), but
  - the linear programming bound shows that MÂ â‰¥ 1142 657488 817576 991675 233913 757847 770281 701964 551079 070743 947785 108476 991312 494592 / 2464 861605 749631 > 2²¹⁸ [i]