MinT - Best Known (178−36, 178, s)-Nets in Base 4

Best Known (178−36, 178, s)-Nets in Base 4

(178−36, 178, 1061)-Net over F₄ — Constructive and digital

Digital (142, 178, 1061)-net over F₄, using

4¹ times duplication [i] based on digital (141, 177, 1061)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 33, 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 (108, 144, 1028)-net over F₄, using
    - trace code for nets [i] based on digital (0, 36, 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]

(178−36, 178, 5363)-Net over F₄ — Digital

Digital (142, 178, 5363)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(178−36, 178, 2263134)-Net in Base 4 — Upper bound on s

There is no (142, 178, 2263135)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 146784 416602 430346 691696 022070 971014 054826 015720 548149 891986 478197 749237 426459 106949 195001 933082 469956 748973 > 4¹⁷⁸ [i]