MinT - Best Known (127−51, 127, s)-Nets in Base 9

Best Known (127−51, 127, s)-Nets in Base 9

(127−51, 127, 344)-Net over F₉ — Constructive and digital

Digital (76, 127, 344)-net over F₉, using

11 times m-reduction [i] based on digital (76, 138, 344)-net over F₉, using
- trace code for nets [i] based on digital (7, 69, 172)-net over F₈₁, using
  - net from sequence [i] based on digital (7, 171)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 7 and N(F) ≥ 172, using
      - a function field by SÃ©mirat [i]

(127−51, 127, 654)-Net over F₉ — Digital

Digital (76, 127, 654)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(127−51, 127, 82004)-Net in Base 9 — Upper bound on s

There is no (76, 127, 82005)-net in base 9, because

1 times m-reduction [i] would yield (76, 126, 82005)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 1 716384 391644 405482 774538 900557 489505 277156 432811 544490 675411 041319 058732 394140 030730 528959 807453 679282 278697 114685 142889 > 9¹²⁶ [i]