MinT - Best Known (166−134, 166, s)-Nets in Base 4

Best Known (166−134, 166, s)-Nets in Base 4

Digital (32, 166, 34)-net over F₄, using

t-expansion [i] based on digital (21, 166, 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]

(32, 166, 43)-net in base 4, using

t-expansion [i] based on (30, 166, 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 (32, 166, 60)-net over F₄, using

t-expansion [i] based on digital (31, 166, 60)-net over F₄, using
- net from sequence [i] based on digital (31, 59)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 31 and N(F) ≥ 60, using
    - Drinfeld modules of rank 1 [i]

There is no (32, 166, 134)-net in base 4, because

46 times m-reduction [i] would yield (32, 120, 134)-net in base 4, but
- extracting embedded orthogonal array [i] would yield OA(4¹²⁰, 134, S₄, 88), but
  - the linear programming bound shows that MÂ â‰¥ 44879 226248 006156 277409 262412 286361 464349 961201 014006 795486 400435 014006 889500 377088 / 21749 419055 > 4¹²⁰ [i]