MinT - Best Known (219−107, 219, s)-Nets in Base 2

Best Known (219−107, 219, s)-Nets in Base 2

(219−107, 219, 57)-Net over F₂ — Constructive and digital

Digital (112, 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]

(219−107, 219, 72)-Net over F₂ — Digital

Digital (112, 219, 72)-net over F₂, using

t-expansion [i] based on digital (110, 219, 72)-net over F₂, using
- net from sequence [i] based on digital (110, 71)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 110 and N(F) ≥ 72, using
    - Drinfeld modules of rank 1 [i]

(219−107, 219, 236)-Net in Base 2 — Upper bound on s

There is no (112, 219, 237)-net in base 2, because

1 times m-reduction [i] would yield (112, 218, 237)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2²¹⁸, 237, S₂, 106), but
  - the linear programming bound shows that MÂ â‰¥ 59311 882667 731407 548267 433191 443187 482001 717729 589259 458427 942548 275200 / 110349 > 2²¹⁸ [i]