MinT - Best Known (243−180, 243, s)-Nets in Base 2

Best Known (243−180, 243, s)-Nets in Base 2

(243−180, 243, 43)-Net over F₂ — Constructive and digital

Digital (63, 243, 43)-net over F₂, using

t-expansion [i] based on digital (59, 243, 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]

(243−180, 243, 44)-Net over F₂ — Digital

Digital (63, 243, 44)-net over F₂, using

t-expansion [i] based on digital (62, 243, 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]

(243−180, 243, 80)-Net in Base 2 — Upper bound on s

There is no (63, 243, 81)-net in base 2, because

8 times m-reduction [i] would yield (63, 235, 81)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2²³⁵, 81, S₂, 3, 172), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 12 367929 453448 690307 083082 505200 429610 792387 278129 332706 851243 409048 338432 / 173 > 2²³⁵ [i]