MinT - Best Known (166, 166+56, s)-Nets in Base 4

Best Known (166, 166+56, s)-Nets in Base 4

(166, 166+56, 548)-Net over F₄ — Constructive and digital

Digital (166, 222, 548)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (5, 33, 17)-net over F₄, using
  - net from sequence [i] based on digital (5, 16)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 5 and N(F) ≥ 17, using
      - an explicitly constructive algebraic function field [i]
2. digital (133, 189, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 63, 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]

(166, 166+56, 648)-Net in Base 4 — Constructive

(166, 222, 648)-net in base 4, using

3 times m-reduction [i] based on (166, 225, 648)-net in base 4, using
- trace code for nets [i] based on (16, 75, 216)-net in base 64, using
  - 2 times m-reduction [i] based on (16, 77, 216)-net in base 64, using
    - base change [i] based on digital (5, 66, 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]

(166, 166+56, 1942)-Net over F₄ — Digital

Digital (166, 222, 1942)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(166, 166+56, 223519)-Net in Base 4 — Upper bound on s

There is no (166, 222, 223520)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 45 431313 566636 516686 430881 061623 939903 913781 433880 517767 906655 790537 360822 420853 855402 702691 870393 497295 277388 206351 420595 573855 154559 > 4²²² [i]