MinT - Best Known (106, 106+102, s)-Nets in Base 2

Best Known (106, 106+102, s)-Nets in Base 2

(106, 106+102, 56)-Net over F₂ — Constructive and digital

Digital (106, 208, 56)-net over F₂, using

t-expansion [i] based on digital (105, 208, 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]

(106, 106+102, 65)-Net over F₂ — Digital

Digital (106, 208, 65)-net over F₂, using

t-expansion [i] based on digital (95, 208, 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]

(106, 106+102, 224)-Net in Base 2 — Upper bound on s

There is no (106, 208, 225)-net in base 2, because

2 times m-reduction [i] would yield (106, 206, 225)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2²⁰⁶, 225, S₂, 100), but
  - the linear programming bound shows that MÂ â‰¥ 46 910043 920112 941849 753861 456344 948196 047834 057482 094744 342146 056192 / 365313 > 2²⁰⁶ [i]