MinT - Best Known (170−34, 170, s)-Nets in Base 4

Best Known (170−34, 170, s)-Nets in Base 4

(170−34, 170, 1062)-Net over F₄ — Constructive and digital

Digital (136, 170, 1062)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (17, 34, 34)-net over F₄, using
  - trace code for nets [i] based on digital (0, 17, 17)-net over F₁₆, using
    - net from sequence [i] based on digital (0, 16)-sequence over F₁₆, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 0 and N(F) ≥ 17, using
        the rational function field F₁₆(x) [i]
      - Niederreiter sequence [i]
2. digital (102, 136, 1028)-net over F₄, using
  - trace code for nets [i] based on digital (0, 34, 257)-net over F₂₅₆, using
    - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
      - Niederreiter sequence [i]

(170−34, 170, 5559)-Net over F₄ — Digital

Digital (136, 170, 5559)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(170−34, 170, 2508542)-Net in Base 4 — Upper bound on s

There is no (136, 170, 2508543)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2 239749 175432 781468 152988 877024 162780 585569 520846 169224 728377 590811 308176 196528 411795 273346 612229 805166 > 4¹⁷⁰ [i]