MinT - Best Known (109−51, 109, s)-Nets in Base 4

Best Known (109−51, 109, s)-Nets in Base 4

(109−51, 109, 76)-Net over F₄ — Constructive and digital

Digital (58, 109, 76)-net over F₄, using

1 times m-reduction [i] based on digital (58, 110, 76)-net over F₄, using
- trace code for nets [i] based on digital (3, 55, 38)-net over F₁₆, using
  - net from sequence [i] based on digital (3, 37)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 3 and N(F) ≥ 38, using
      - a function field by SÃ©mirat [i]

(109−51, 109, 106)-Net over F₄ — Digital

Digital (58, 109, 106)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(109−51, 109, 1333)-Net in Base 4 — Upper bound on s

There is no (58, 109, 1334)-net in base 4, because

1 times m-reduction [i] would yield (58, 108, 1334)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 107050 562901 309070 997164 671263 301023 182221 788966 404583 119100 130988 > 4¹⁰⁸ [i]