MinT - Best Known (185−61, 185, s)-Nets in Base 4

Best Known (185−61, 185, s)-Nets in Base 4

(185−61, 185, 195)-Net over F₄ — Constructive and digital

Digital (124, 185, 195)-net over F₄, using

1 times m-reduction [i] based on digital (124, 186, 195)-net over F₄, using
- trace code for nets [i] based on digital (0, 62, 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]

(185−61, 185, 240)-Net in Base 4 — Constructive

(124, 185, 240)-net in base 4, using

3 times m-reduction [i] based on (124, 188, 240)-net in base 4, using
- trace code for nets [i] based on (30, 94, 120)-net in base 16, using
  - 1 times m-reduction [i] based on (30, 95, 120)-net in base 16, using
    - base change [i] based on digital (11, 76, 120)-net over F₃₂, using
      - net from sequence [i] based on digital (11, 119)-sequence over F₃₂, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 11 and N(F) ≥ 120, using
        a function field by SÃ©mirat [i]

(185−61, 185, 535)-Net over F₄ — Digital

Digital (124, 185, 535)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(185−61, 185, 19758)-Net in Base 4 — Upper bound on s

There is no (124, 185, 19759)-net in base 4, because

1 times m-reduction [i] would yield (124, 184, 19759)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 601 261932 395834 079866 535818 272745 531469 270616 506348 884050 575725 583464 212202 623847 743499 490695 862966 401705 088298 > 4¹⁸⁴ [i]