MinT - Best Known (30, 30+79, s)-Nets in Base 4

Best Known (30, 30+79, s)-Nets in Base 4

Digital (30, 109, 34)-net over F₄, using

t-expansion [i] based on digital (21, 109, 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, 109, 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, 109, 55)-net over F₄, using

t-expansion [i] based on digital (26, 109, 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, 109, 129)-net in base 4, because

extracting embedded orthogonal array [i] would yield OA(4¹⁰⁹, 129, S₄, 79), but
- the linear programming bound shows that MÂ â‰¥ 1283 538057 535267 617802 716786 606074 064002 460629 607029 943501 790942 040867 039548 014592 / 2304 685234 955125 > 4¹⁰⁹ [i]