MinT - Best Known (210−71, 210, s)-Nets in Base 4

Best Known (210−71, 210, s)-Nets in Base 4

(210−71, 210, 164)-Net over F₄ — Constructive and digital

Digital (139, 210, 164)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (21, 56, 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]
2. digital (83, 154, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 77, 65)-net over F₁₆, using
    - net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
        the Hermitian function field over F₁₆ [i]

(210−71, 210, 240)-Net in Base 4 — Constructive

(139, 210, 240)-net in base 4, using

t-expansion [i] based on (137, 210, 240)-net in base 4, using
- trace code for nets [i] based on (32, 105, 120)-net in base 16, using
  - base change [i] based on digital (11, 84, 120)-net over F₃₂, using
    - net from sequence [i] based on digital (11, 119)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 11 and N(F) ≥ 120, using
        a function field by SÃ©mirat [i]

(210−71, 210, 545)-Net over F₄ — Digital

Digital (139, 210, 545)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(210−71, 210, 18222)-Net in Base 4 — Upper bound on s

There is no (139, 210, 18223)-net in base 4, because

1 times m-reduction [i] would yield (139, 209, 18223)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 676994 882069 920241 589533 685654 381104 236942 954935 745858 085689 671647 492537 215294 949488 672000 038607 092434 404907 312024 880361 044808 > 4²⁰⁹ [i]