MinT - Best Known (166−57, 166, s)-Nets in Base 4

Best Known (166−57, 166, s)-Nets in Base 4

(166−57, 166, 158)-Net over F₄ — Constructive and digital

Digital (109, 166, 158)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (12, 40, 28)-net over F₄, using
  - net from sequence [i] based on digital (12, 27)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 12 and N(F) ≥ 28, using
      - an explicitly constructive algebraic function field [i]
2. digital (69, 126, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 63, 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]

(166−57, 166, 208)-Net in Base 4 — Constructive

(109, 166, 208)-net in base 4, using

trace code for nets [i] based on (26, 83, 104)-net in base 16, using
- 2 times m-reduction [i] based on (26, 85, 104)-net in base 16, using
  - base change [i] based on digital (9, 68, 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]

(166−57, 166, 420)-Net over F₄ — Digital

Digital (109, 166, 420)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(166−57, 166, 13273)-Net in Base 4 — Upper bound on s

There is no (109, 166, 13274)-net in base 4, because

1 times m-reduction [i] would yield (109, 165, 13274)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2188 723141 091619 410004 780949 189550 406370 245157 543586 422861 239357 613020 153354 633863 920676 481253 425328 > 4¹⁶⁵ [i]