MinT - Best Known (31, 68, s)-Nets in Base 4

Best Known (31, 68, s)-Nets in Base 4

Digital (31, 68, 41)-net over F₄, using

(31, 68, 43)-net in base 4, using

t-expansion [i] based on (30, 68, 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, 68, 60)-net over F₄, using

There is no (31, 68, 425)-net in base 4, because

1 times m-reduction [i] would yield (31, 67, 425)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 22483 131241 189049 358091 742416 208617 277416 > 4⁶⁷ [i]