MinT - Best Known (125−57, 125, s)-Nets in Base 4

Best Known (125−57, 125, s)-Nets in Base 4

(125−57, 125, 98)-Net over F₄ — Constructive and digital

Digital (68, 125, 98)-net over F₄, using

1 times m-reduction [i] based on digital (68, 126, 98)-net over F₄, using
- trace code for nets [i] based on digital (5, 63, 49)-net over F₁₆, using
  - net from sequence [i] based on digital (5, 48)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 5 and N(F) ≥ 49, using
      - a function field by SÃ©mirat [i]

(125−57, 125, 129)-Net over F₄ — Digital

Digital (68, 125, 129)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(125−57, 125, 1723)-Net in Base 4 — Upper bound on s

There is no (68, 125, 1724)-net in base 4, because

1 times m-reduction [i] would yield (68, 124, 1724)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 454 465968 246032 539973 502112 720322 269875 216723 076803 380048 221046 887982 727256 > 4¹²⁴ [i]