MinT - Best Known (220−153, 220, s)-Nets in Base 2

Best Known (220−153, 220, s)-Nets in Base 2

(220−153, 220, 43)-Net over F₂ — Constructive and digital

Digital (67, 220, 43)-net over F₂, using

t-expansion [i] based on digital (59, 220, 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]

(220−153, 220, 48)-Net over F₂ — Digital

Digital (67, 220, 48)-net over F₂, using

t-expansion [i] based on digital (65, 220, 48)-net over F₂, using
- net from sequence [i] based on digital (65, 47)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 65 and N(F) ≥ 48, using
    - Drinfeld modules of rank 1 [i]

(220−153, 220, 98)-Net in Base 2 — Upper bound on s

There is no (67, 220, 99)-net in base 2, because

29 times m-reduction [i] would yield (67, 191, 99)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2¹⁹¹, 99, S₂, 2, 124), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 100433 627766 186892 221372 630771 322662 657637 687111 424552 206336 / 25 > 2¹⁹¹ [i]