MinT - Best Known (109, 109+122, s)-Nets in Base 2

Best Known (109, 109+122, s)-Nets in Base 2

(109, 109+122, 56)-Net over F₂ — Constructive and digital

Digital (109, 231, 56)-net over F₂, using

t-expansion [i] based on digital (105, 231, 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]

(109, 109+122, 65)-Net over F₂ — Digital

Digital (109, 231, 65)-net over F₂, using

t-expansion [i] based on digital (95, 231, 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]

(109, 109+122, 229)-Net in Base 2 — Upper bound on s

There is no (109, 231, 230)-net in base 2, because

14 times m-reduction [i] would yield (109, 217, 230)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2²¹⁷, 230, S₂, 108), but
  - the linear programming bound shows that MÂ â‰¥ 26 959946 667150 639794 667015 087019 630673 637144 422540 572481 103610 249216 / 93 > 2²¹⁷ [i]