MinT - Best Known (227−120, 227, s)-Nets in Base 2

Best Known (227−120, 227, s)-Nets in Base 2

(227−120, 227, 56)-Net over F₂ — Constructive and digital

Digital (107, 227, 56)-net over F₂, using

t-expansion [i] based on digital (105, 227, 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]

(227−120, 227, 65)-Net over F₂ — Digital

Digital (107, 227, 65)-net over F₂, using

t-expansion [i] based on digital (95, 227, 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]

(227−120, 227, 224)-Net over F₂ — Upper bound on s (digital)

There is no digital (107, 227, 225)-net over F₂, because

8 times m-reduction [i] would yield digital (107, 219, 225)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2²¹⁹, 225, F₂, 112) (dual of [225, 6, 113]-code), but
  - Griesmer bound [i]

(227−120, 227, 225)-Net in Base 2 — Upper bound on s

There is no (107, 227, 226)-net in base 2, because

14 times m-reduction [i] would yield (107, 213, 226)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2²¹³, 226, S₂, 106), but
  - the linear programming bound shows that MÂ â‰¥ 309 196888 338883 900145 087329 279256 389288 276000 096012 190642 657030 045696 / 17019 > 2²¹³ [i]