MinT - Best Known (208−53, 208, s)-Nets in Base 4

Best Known (208−53, 208, s)-Nets in Base 4

(208−53, 208, 541)-Net over F₄ — Constructive and digital

Digital (155, 208, 541)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (2, 28, 10)-net over F₄, using
  - net from sequence [i] based on digital (2, 9)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 2 and N(F) ≥ 10, using
      - an explicitly constructive algebraic function field [i]
2. digital (127, 180, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 60, 177)-net over F₆₄, using
    - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
        a function field by GarcÃa and Quoos [i]

(208−53, 208, 648)-Net in Base 4 — Constructive

(155, 208, 648)-net in base 4, using

2 times m-reduction [i] based on (155, 210, 648)-net in base 4, using
- trace code for nets [i] based on (15, 70, 216)-net in base 64, using
  - base change [i] based on digital (5, 60, 216)-net over F₁₂₈, using
    - net from sequence [i] based on digital (5, 215)-sequence over F₁₂₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 5 and N(F) ≥ 216, using
        a function field by SÃ©mirat [i]

(208−53, 208, 1752)-Net over F₄ — Digital

Digital (155, 208, 1752)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(208−53, 208, 218501)-Net in Base 4 — Upper bound on s

There is no (155, 208, 218502)-net in base 4, because

1 times m-reduction [i] would yield (155, 207, 218502)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 42312 450117 878101 355089 139592 027078 850412 558280 130025 142256 892161 762448 961874 637820 327807 772860 892601 010346 536094 447225 405408 > 4²⁰⁷ [i]