MinT - Best Known (165−134, 165, s)-Nets in Base 4

Best Known (165−134, 165, s)-Nets in Base 4

Digital (31, 165, 34)-net over F₄, using

t-expansion [i] based on digital (21, 165, 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]

(31, 165, 43)-net in base 4, using

t-expansion [i] based on (30, 165, 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 (31, 165, 60)-net over F₄, using

There is no (31, 165, 130)-net in base 4, because

49 times m-reduction [i] would yield (31, 116, 130)-net in base 4, but
- extracting embedded orthogonal array [i] would yield OA(4¹¹⁶, 130, S₄, 85), but
  - the linear programming bound shows that MÂ â‰¥ 2910 853909 303427 843179 867153 014027 057557 909168 265288 524176 409407 780925 198594 736128 / 395256 842585 > 4¹¹⁶ [i]