MinT - Best Known (137−45, 137, s)-Nets in Base 4

Best Known (137−45, 137, s)-Nets in Base 4

(137−45, 137, 195)-Net over F₄ — Constructive and digital

Digital (92, 137, 195)-net over F₄, using

1 times m-reduction [i] based on digital (92, 138, 195)-net over F₄, using
- trace code for nets [i] based on digital (0, 46, 65)-net over F₆₄, using
  - net from sequence [i] based on digital (0, 64)-sequence over F₆₄, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 0 and N(F) ≥ 65, using
      - the rational function field F₆₄(x) [i]
    - Niederreiter sequence [i]

(137−45, 137, 208)-Net in Base 4 — Constructive

(92, 137, 208)-net in base 4, using

1 times m-reduction [i] based on (92, 138, 208)-net in base 4, using
- trace code for nets [i] based on (23, 69, 104)-net in base 16, using
  - 1 times m-reduction [i] based on (23, 70, 104)-net in base 16, using
    - base change [i] based on digital (9, 56, 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]

(137−45, 137, 423)-Net over F₄ — Digital

Digital (92, 137, 423)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(137−45, 137, 15888)-Net in Base 4 — Upper bound on s

There is no (92, 137, 15889)-net in base 4, because

1 times m-reduction [i] would yield (92, 136, 15889)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 7598 685199 461135 498163 808442 247747 646902 473489 704403 665509 399305 621495 963476 320000 > 4¹³⁶ [i]