MinT - Best Known (239−63, 239, s)-Nets in Base 4

Best Known (239−63, 239, s)-Nets in Base 4

(239−63, 239, 531)-Net over F₄ — Constructive and digital

Digital (176, 239, 531)-net over F₄, using

t-expansion [i] based on digital (175, 239, 531)-net over F₄, using
- 13 times m-reduction [i] based on digital (175, 252, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 84, 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]

(239−63, 239, 576)-Net in Base 4 — Constructive

(176, 239, 576)-net in base 4, using

t-expansion [i] based on (175, 239, 576)-net in base 4, using
- 1 times m-reduction [i] based on (175, 240, 576)-net in base 4, using
  - trace code for nets [i] based on (15, 80, 192)-net in base 64, using
    - 4 times m-reduction [i] based on (15, 84, 192)-net in base 64, using
      - base change [i] based on digital (3, 72, 192)-net over F₁₂₈, using
        net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 3 and N(F) ≥ 192, using
        a function field by SÃ©mirat [i]

(239−63, 239, 1701)-Net over F₄ — Digital

Digital (176, 239, 1701)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(239−63, 239, 173426)-Net in Base 4 — Upper bound on s

There is no (176, 239, 173427)-net in base 4, because

1 times m-reduction [i] would yield (176, 238, 173427)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 195128 882789 624723 633567 937649 194392 600138 599388 850086 761287 867802 855927 361584 314510 172415 428114 836226 649695 840867 810548 206974 184133 167787 431680 > 4²³⁸ [i]