MinT - Best Known (219−189, 219, s)-Nets in Base 4

Best Known (219−189, 219, s)-Nets in Base 4

Digital (30, 219, 34)-net over F₄, using

t-expansion [i] based on digital (21, 219, 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]

(30, 219, 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, 219, 55)-net over F₄, using

t-expansion [i] based on digital (26, 219, 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, 219, 108)-net in base 4, because

7 times m-reduction [i] would yield (30, 212, 108)-net in base 4, but
- extracting embedded OOA [i] would yield OOA(4²¹², 108, S₄, 2, 182), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 2772 669694 120814 859578 414184 143083 703436 437075 375816 575170 479580 614621 307805 625623 039974 406104 139578 097391 210961 403571 828974 157824 / 61 > 4²¹² [i]