MinT - Best Known (212−187, 212, s)-Nets in Base 4

Best Known (212−187, 212, s)-Nets in Base 4

Digital (25, 212, 34)-net over F₄, using

t-expansion [i] based on digital (21, 212, 34)-net over F₄, using
- net from sequence [i] based on digital (21, 33)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 21 and N(F) ≥ 34, using
    - T₅ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(25, 212, 35)-net in base 4, using

t-expansion [i] based on (24, 212, 35)-net in base 4, using
- net from sequence [i] based on (24, 34)-sequence in base 4, using
  - base expansion [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]

Digital (25, 212, 51)-net over F₄, using

There is no (25, 212, 92)-net in base 4, because

32 times m-reduction [i] would yield (25, 180, 92)-net in base 4, but
- extracting embedded OOA [i] would yield OOA(4¹⁸⁰, 92, S₄, 2, 155), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 37 576681 324381 331646 231689 548629 392438 010920 782533 117931 316655 544515 344401 833735 095419 183974 156299 248510 959616 / 13 > 4¹⁸⁰ [i]