MinT - Best Known (109−57, 109, s)-Nets in Base 2

Best Known (109−57, 109, s)-Nets in Base 2

(109−57, 109, 36)-Net over F₂ — Constructive and digital

Digital (52, 109, 36)-net over F₂, using

t-expansion [i] based on digital (51, 109, 36)-net over F₂, using
- net from sequence [i] based on digital (51, 35)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 41, N(F)Â =Â 32, 1 place with degree 2, and 3 places with degree 4 [i] based on function field F/F₂ with g(F) = 41 and N(F) ≥ 32, using an explicitly constructive algebraic function field [i]

(109−57, 109, 40)-Net over F₂ — Digital

Digital (52, 109, 40)-net over F₂, using

t-expansion [i] based on digital (50, 109, 40)-net over F₂, using
- net from sequence [i] based on digital (50, 39)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 50 and N(F) ≥ 40, using
    - Drinfeld modules of rank 1 [i]

(109−57, 109, 113)-Net over F₂ — Upper bound on s (digital)

There is no digital (52, 109, 114)-net over F₂, because

1 times m-reduction [i] would yield digital (52, 108, 114)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2¹⁰⁸, 114, F₂, 56) (dual of [114, 6, 57]-code), but
  - Griesmer bound [i]

(109−57, 109, 115)-Net in Base 2 — Upper bound on s

There is no (52, 109, 116)-net in base 2, because

5 times m-reduction [i] would yield (52, 104, 116)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2¹⁰⁴, 116, S₂, 52), but
  - the linear programming bound shows that MÂ â‰¥ 2271 629875 608987 087482 092144 033792 / 99 > 2¹⁰⁴ [i]