MinT - Best Known (138, 138+53, s)-Nets in Base 4

Best Known (138, 138+53, s)-Nets in Base 4

(138, 138+53, 531)-Net over F₄ — Constructive and digital

Digital (138, 191, 531)-net over F₄, using

t-expansion [i] based on digital (137, 191, 531)-net over F₄, using
- 4 times m-reduction [i] based on digital (137, 195, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 65, 177)-net over F₆₄, using
    - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
        a function field by GarcÃa and Quoos [i]

(138, 138+53, 1120)-Net over F₄ — Digital

Digital (138, 191, 1120)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(138, 138+53, 88254)-Net in Base 4 — Upper bound on s

There is no (138, 191, 88255)-net in base 4, because

1 times m-reduction [i] would yield (138, 190, 88255)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2 462994 605538 956164 932079 192906 484782 363659 642617 092656 985879 387537 836491 473196 866713 860855 268424 083393 572056 534399 > 4¹⁹⁰ [i]