MinT - Best Known (187−123, 187, s)-Nets in Base 2

Best Known (187−123, 187, s)-Nets in Base 2

(187−123, 187, 43)-Net over F₂ — Constructive and digital

Digital (64, 187, 43)-net over F₂, using

t-expansion [i] based on digital (59, 187, 43)-net over F₂, using
- net from sequence [i] based on digital (59, 42)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 54, N(F)Â =Â 42, and 1 place with degree 6 [i] based on function field F/F₂ with g(F) = 54 and N(F) ≥ 42, using an explicitly constructive algebraic function field [i]

(187−123, 187, 44)-Net over F₂ — Digital

Digital (64, 187, 44)-net over F₂, using

t-expansion [i] based on digital (62, 187, 44)-net over F₂, using
- net from sequence [i] based on digital (62, 43)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 62 and N(F) ≥ 44, using
    - a function field from a table by Niederreiter and Xing [i]

(187−123, 187, 94)-Net in Base 2 — Upper bound on s

There is no (64, 187, 95)-net in base 2, because

4 times m-reduction [i] would yield (64, 183, 95)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2¹⁸³, 95, S₂, 2, 119), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 49 039857 307708 443467 467104 868809 893875 799651 909875 269632 / 3 > 2¹⁸³ [i]