MinT - Best Known (138−112, 138, s)-Nets in Base 4

Best Known (138−112, 138, s)-Nets in Base 4

Digital (26, 138, 34)-net over F₄, using

t-expansion [i] based on digital (21, 138, 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]

(26, 138, 36)-net in base 4, using

net from sequence [i] based on (26, 35)-sequence in base 4, using
- base expansion [i] based on digital (52, 35)-sequence over F₂, using
  - t-expansion [i] based on digital (51, 35)-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 3 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 (26, 138, 55)-net over F₄, using

There is no (26, 138, 112)-net in base 4, because

40 times m-reduction [i] would yield (26, 98, 112)-net in base 4, but
- extracting embedded orthogonal array [i] would yield OA(4⁹⁸, 112, S₄, 72), but
  - the linear programming bound shows that MÂ â‰¥ 1 166662 731140 735083 887873 150433 161508 032759 728734 208123 114163 798016 / 9 452405 > 4⁹⁸ [i]