MinT - Best Known (109−84, 109, s)-Nets in Base 4

Best Known (109−84, 109, s)-Nets in Base 4

Digital (25, 109, 34)-net over F₄, using

t-expansion [i] based on digital (21, 109, 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]

(25, 109, 35)-net in base 4, using

t-expansion [i] based on (24, 109, 35)-net in base 4, using
- net from sequence [i] based on (24, 34)-sequence in base 4, using
  - base expansion [i] based on digital (48, 34)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 41, N(F)Â =Â 32, 1 place with degree 2, and 2 places with degree 4 [i] based on function field F/F₂ with g(F) = 41 and N(F) ≥ 32, using an explicitly constructive algebraic function field [i]

Digital (25, 109, 51)-net over F₄, using

There is no (25, 109, 108)-net in base 4, because

12 times m-reduction [i] would yield (25, 97, 108)-net in base 4, but
- extracting embedded orthogonal array [i] would yield OA(4⁹⁷, 108, S₄, 72), but
  - the linear programming bound shows that MÂ â‰¥ 9230 252126 223640 142713 030258 407637 988887 533996 288362 045971 103744 / 357335 > 4⁹⁷ [i]