MinT - Best Known (194−55, 194, s)-Nets in Base 4

Best Known (194−55, 194, s)-Nets in Base 4

(194−55, 194, 531)-Net over F₄ — Constructive and digital

Digital (139, 194, 531)-net over F₄, using

4 times m-reduction [i] based on digital (139, 198, 531)-net over F₄, using
- trace code for nets [i] based on digital (7, 66, 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]

(194−55, 194, 1031)-Net over F₄ — Digital

Digital (139, 194, 1031)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(194−55, 194, 73243)-Net in Base 4 — Upper bound on s

There is no (139, 194, 73244)-net in base 4, because

1 times m-reduction [i] would yield (139, 193, 73244)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 157 636826 572511 692417 976977 788981 457786 029427 761570 652681 318498 582512 497872 266033 795124 588718 227657 901183 113188 064975 > 4¹⁹³ [i]