MinT - Best Known (32, 32+81, s)-Nets in Base 4

Best Known (32, 32+81, s)-Nets in Base 4

Digital (32, 113, 34)-net over F₄, using

t-expansion [i] based on digital (21, 113, 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, 113, 43)-net in base 4, using

t-expansion [i] based on (30, 113, 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, 113, 60)-net over F₄, using

t-expansion [i] based on digital (31, 113, 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, 113, 139)-net in base 4, because

extracting embedded orthogonal array [i] would yield OA(4¹¹³, 139, S₄, 81), but
- the linear programming bound shows that MÂ â‰¥ 270 676957 998568 775166 883930 110227 193694 274801 571525 181006 240705 968272 390265 652071 366656 / 2 060995 576092 406625 > 4¹¹³ [i]