MinT - Best Known (81−51, 81, s)-Nets in Base 4

Best Known (81−51, 81, s)-Nets in Base 4

Digital (30, 81, 34)-net over F₄, using

(30, 81, 43)-net in base 4, using

net from sequence [i] based on (30, 42)-sequence in base 4, using
- base expansion [i] based on digital (60, 42)-sequence over F₂, using
  - t-expansion [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]

Digital (30, 81, 55)-net over F₄, using

t-expansion [i] based on digital (26, 81, 55)-net over F₄, using
- net from sequence [i] based on digital (26, 54)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 26 and N(F) ≥ 55, using
    - Drinfeld modules of rank 1 [i]

There is no (30, 81, 267)-net in base 4, because

1 times m-reduction [i] would yield (30, 80, 267)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1 502308 656899 376462 803270 228429 450416 540565 633824 > 4⁸⁰ [i]