MinT - Best Known (89−29, 89, s)-Nets in Base 4

Best Known (89−29, 89, s)-Nets in Base 4

(89−29, 89, 195)-Net over F₄ — Constructive and digital

Digital (60, 89, 195)-net over F₄, using

1 times m-reduction [i] based on digital (60, 90, 195)-net over F₄, using
- trace code for nets [i] based on digital (0, 30, 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]

(89−29, 89, 316)-Net over F₄ — Digital

Digital (60, 89, 316)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(89−29, 89, 12255)-Net in Base 4 — Upper bound on s

There is no (60, 89, 12256)-net in base 4, because

1 times m-reduction [i] would yield (60, 88, 12256)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 95875 352540 128708 693398 832304 682837 242991 307835 560645 > 4⁸⁸ [i]