MinT - Best Known (109−32, 109, s)-Nets in Base 4

Best Known (109−32, 109, s)-Nets in Base 4

(109−32, 109, 312)-Net over F₄ — Constructive and digital

Digital (77, 109, 312)-net over F₄, using

2 times m-reduction [i] based on digital (77, 111, 312)-net over F₄, using
- trace code for nets [i] based on digital (3, 37, 104)-net over F₆₄, using
  - net from sequence [i] based on digital (3, 103)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 3 and N(F) ≥ 104, using
      - a function field by SÃ©mirat [i]

(109−32, 109, 557)-Net over F₄ — Digital

Digital (77, 109, 557)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(109−32, 109, 28625)-Net in Base 4 — Upper bound on s

There is no (77, 109, 28626)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 421313 225835 191048 083357 354563 874550 597169 794241 213516 601377 154021 > 4¹⁰⁹ [i]