MinT - Best Known (135, 135+34, s)-Nets in Base 4

Best Known (135, 135+34, s)-Nets in Base 4

(135, 135+34, 1061)-Net over F₄ — Constructive and digital

Digital (135, 169, 1061)-net over F₄, using

4¹ times duplication [i] based on digital (134, 168, 1061)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 32, 33)-net over F₄, using
    - net from sequence [i] based on digital (15, 32)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 15 and N(F) ≥ 33, using
        an explicitly constructive algebraic function field [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]

(135, 135+34, 5331)-Net over F₄ — Digital

Digital (135, 169, 5331)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(135, 135+34, 2312096)-Net in Base 4 — Upper bound on s

There is no (135, 169, 2312097)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 559937 183861 062238 030710 571298 737957 037272 706609 425564 646680 646157 203595 335123 181821 316777 461205 462360 > 4¹⁶⁹ [i]