MinT - Best Known (257−221, 257, s)-Nets in Base 2

Best Known (257−221, 257, s)-Nets in Base 2

(257−221, 257, 24)-Net over F₂ — Constructive and digital

Digital (36, 257, 24)-net over F₂, using

t-expansion [i] based on digital (33, 257, 24)-net over F₂, using
- net from sequence [i] based on digital (33, 23)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 33 and N(F) ≥ 24, using
    - a function field by SÃ©mirat [i]

(257−221, 257, 30)-Net over F₂ — Digital

Digital (36, 257, 30)-net over F₂, using

net from sequence [i] based on digital (36, 29)-sequence over F₂, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 36 and N(F) ≥ 30, using
  - Drinfeld modules of rank 1 [i]

(257−221, 257, 44)-Net in Base 2 — Upper bound on s

There is no (36, 257, 45)-net in base 2, because

39 times m-reduction [i] would yield (36, 218, 45)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2²¹⁸, 45, S₂, 5, 182), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 28 644943 333847 554781 833703 529958 357590 739465 948949 358261 172585 889792 / 61 > 2²¹⁸ [i]