MinT - Best Known (31, 113, s)-Nets in Base 4

Best Known (31, 113, s)-Nets in Base 4

Digital (31, 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]

(31, 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 (31, 113, 60)-net over F₄, using

There is no (31, 113, 132)-net in base 4, because

1 times m-reduction [i] would yield (31, 112, 132)-net in base 4, but
- extracting embedded orthogonal array [i] would yield OA(4¹¹², 132, S₄, 81), but
  - the linear programming bound shows that MÂ â‰¥ 11778 504777 779688 498524 359999 043750 103927 634385 012592 570847 246118 930944 386301 689856 / 405 533359 108881 > 4¹¹² [i]