MinT - Best Known (247−89, 247, s)-Nets in Base 4

Best Known (247−89, 247, s)-Nets in Base 4

(247−89, 247, 160)-Net over F₄ — Constructive and digital

Digital (158, 247, 160)-net over F₄, using

t-expansion [i] based on digital (157, 247, 160)-net over F₄, using
- 12 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]

(247−89, 247, 208)-Net in Base 4 — Constructive

(158, 247, 208)-net in base 4, using

1 times m-reduction [i] based on (158, 248, 208)-net in base 4, using
- trace code for nets [i] based on (34, 124, 104)-net in base 16, using
  - 1 times m-reduction [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]

(247−89, 247, 503)-Net over F₄ — Digital

Digital (158, 247, 503)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(247−89, 247, 13325)-Net in Base 4 — Upper bound on s

There is no (158, 247, 13326)-net in base 4, because

1 times m-reduction [i] would yield (158, 246, 13326)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 12801 628684 051097 306102 119161 333858 092906 388533 573383 555844 167314 876615 169289 610554 978719 195206 602732 599128 714624 017449 528746 400436 711366 777968 696672 > 4²⁴⁶ [i]