MinT - Best Known (226−178, 226, s)-Nets in Base 2

Best Known (226−178, 226, s)-Nets in Base 2

(226−178, 226, 35)-Net over F₂ — Constructive and digital

Digital (48, 226, 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]

(226−178, 226, 36)-Net over F₂ — Digital

Digital (48, 226, 36)-net over F₂, using

t-expansion [i] based on digital (47, 226, 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]

(226−178, 226, 62)-Net in Base 2 — Upper bound on s

There is no (48, 226, 63)-net in base 2, because

44 times m-reduction [i] would yield (48, 182, 63)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2¹⁸², 63, S₂, 3, 134), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 98 079714 615416 886934 934209 737619 787751 599303 819750 539264 / 15 > 2¹⁸² [i]