MinT - Best Known (127−31, 127, s)-Nets in Base 4

Best Known (127−31, 127, s)-Nets in Base 4

(127−31, 127, 1028)-Net over F₄ — Constructive and digital

Digital (96, 127, 1028)-net over F₄, using

1 times m-reduction [i] based on digital (96, 128, 1028)-net over F₄, using
- trace code for nets [i] based on digital (0, 32, 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]

(127−31, 127, 1435)-Net over F₄ — Digital

Digital (96, 127, 1435)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(127−31, 127, 244302)-Net in Base 4 — Upper bound on s

There is no (96, 127, 244303)-net in base 4, because

1 times m-reduction [i] would yield (96, 126, 244303)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 7237 210099 361369 677874 556743 240567 922880 734711 201601 183714 209231 148269 569884 > 4¹²⁶ [i]