MinT - Best Known (126−31, 126, s)-Nets in Base 4

Best Known (126−31, 126, s)-Nets in Base 4

(126−31, 126, 1028)-Net over F₄ — Constructive and digital

Digital (95, 126, 1028)-net over F₄, using

4² times duplication [i] based on digital (93, 124, 1028)-net over F₄, using
- trace code for nets [i] based on digital (0, 31, 257)-net over F₂₅₆, using
  - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
      - the rational function field F₂₅₆(x) [i]
    - Niederreiter sequence [i]

(126−31, 126, 1371)-Net over F₄ — Digital

Digital (95, 126, 1371)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(126−31, 126, 222735)-Net in Base 4 — Upper bound on s

There is no (95, 126, 222736)-net in base 4, because

1 times m-reduction [i] would yield (95, 125, 222736)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1809 365902 802546 129560 998660 104935 570645 392365 789774 651944 993258 084157 534335 > 4¹²⁵ [i]