MinT - Best Known (231−168, 231, s)-Nets in Base 2

Best Known (231−168, 231, s)-Nets in Base 2

(231−168, 231, 43)-Net over F₂ — Constructive and digital

Digital (63, 231, 43)-net over F₂, using

t-expansion [i] based on digital (59, 231, 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]

(231−168, 231, 44)-Net over F₂ — Digital

Digital (63, 231, 44)-net over F₂, using

t-expansion [i] based on digital (62, 231, 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]

(231−168, 231, 93)-Net in Base 2 — Upper bound on s

There is no (63, 231, 94)-net in base 2, because

51 times m-reduction [i] would yield (63, 180, 94)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2¹⁸⁰, 94, S₂, 2, 117), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 98 079714 615416 886934 934209 737619 787751 599303 819750 539264 / 59 > 2¹⁸⁰ [i]