MinT - Best Known (250−91, 250, s)-Nets in Base 4

Best Known (250−91, 250, s)-Nets in Base 4

(250−91, 250, 160)-Net over F₄ — Constructive and digital

Digital (159, 250, 160)-net over F₄, using

t-expansion [i] based on digital (157, 250, 160)-net over F₄, using
- 9 times m-reduction [i] based on digital (157, 259, 160)-net over F₄, using
  - (u,Â u+v)-construction [i] based on
    1. digital (33, 84, 56)-net over F₄, using
      - net from sequence [i] based on digital (33, 55)-sequence over F₄, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 33 and N(F) ≥ 56, using
        F₅ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]
    2. digital (73, 175, 104)-net over F₄, using
      - net from sequence [i] based on digital (73, 103)-sequence over F₄, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 73 and N(F) ≥ 104, using
        F₆ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(250−91, 250, 208)-Net in Base 4 — Constructive

(159, 250, 208)-net in base 4, using

trace code for nets [i] based on (34, 125, 104)-net in base 16, using
- base change [i] based on digital (9, 100, 104)-net over F₃₂, using
  - net from sequence [i] based on digital (9, 103)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 9 and N(F) ≥ 104, using
      - a function field by SÃ©mirat [i]

(250−91, 250, 490)-Net over F₄ — Digital

Digital (159, 250, 490)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(250−91, 250, 12565)-Net in Base 4 — Upper bound on s

There is no (159, 250, 12566)-net in base 4, because

1 times m-reduction [i] would yield (159, 249, 12566)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 820176 498014 994944 502036 762987 733679 338574 829930 474365 233264 799100 735505 806740 871429 374780 930110 511176 007559 226779 423796 003664 660703 918684 687598 312720 > 4²⁴⁹ [i]