MinT - Best Known (49, 253, s)-Nets in Base 2

Best Known (49, 253, s)-Nets in Base 2

(49, 253, 35)-Net over F₂ — Constructive and digital

Digital (49, 253, 35)-net over F₂, using

t-expansion [i] based on digital (48, 253, 35)-net over F₂, using
- net from sequence [i] based on digital (48, 34)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 41, N(F)Â =Â 32, 1 place with degree 2, and 2 places with degree 4 [i] based on function field F/F₂ with g(F) = 41 and N(F) ≥ 32, using an explicitly constructive algebraic function field [i]

(49, 253, 36)-Net over F₂ — Digital

Digital (49, 253, 36)-net over F₂, using

t-expansion [i] based on digital (47, 253, 36)-net over F₂, using
- net from sequence [i] based on digital (47, 35)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 47 and N(F) ≥ 36, using
    - Drinfeld modules of rank 1 [i]

(49, 253, 60)-Net in Base 2 — Upper bound on s

There is no (49, 253, 61)-net in base 2, because

17 times m-reduction [i] would yield (49, 236, 61)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2²³⁶, 61, S₂, 4, 187), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 8 392523 557697 325565 520663 128528 862950 180548 510159 190051 077629 456139 943936 / 47 > 2²³⁶ [i]