MinT - Best Known (173−35, 173, s)-Nets in Base 4

Best Known (173−35, 173, s)-Nets in Base 4

(173−35, 173, 1061)-Net over F₄ — Constructive and digital

Digital (138, 173, 1061)-net over F₄, using

4¹ times duplication [i] based on digital (137, 172, 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 (105, 140, 1028)-net over F₄, using
    - trace code for nets [i] based on digital (0, 35, 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]

(173−35, 173, 5238)-Net over F₄ — Digital

Digital (138, 173, 5238)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(173−35, 173, 2952925)-Net in Base 4 — Upper bound on s

There is no (138, 173, 2952926)-net in base 4, because

1 times m-reduction [i] would yield (138, 172, 2952926)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 35 836085 839795 969436 770916 856691 466764 445376 959857 719667 702531 880058 937053 260428 129612 687360 949324 179549 > 4¹⁷² [i]