MinT - Best Known (115−69, 115, s)-Nets in Base 9

Best Known (115−69, 115, s)-Nets in Base 9

(115−69, 115, 81)-Net over F₉ — Constructive and digital

Digital (46, 115, 81)-net over F₉, using

t-expansion [i] based on digital (32, 115, 81)-net over F₉, using
- net from sequence [i] based on digital (32, 80)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 32 and N(F) ≥ 81, using
    - F₄ from the tower of function fields by Bezerra and GarcÃa over F₉ [i]

(115−69, 115, 82)-Net in Base 9 — Constructive

(46, 115, 82)-net in base 9, using

2 times m-reduction [i] based on (46, 117, 82)-net in base 9, using
- base change [i] based on digital (7, 78, 82)-net over F₂₇, using
  - net from sequence [i] based on digital (7, 81)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 7 and N(F) ≥ 82, using
      - a function field by SÃ©mirat [i]

(115−69, 115, 162)-Net over F₉ — Digital

Digital (46, 115, 162)-net over F₉, using

net from sequence [i] based on digital (46, 161)-sequence over F₉, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 46 and N(F) ≥ 162, using
  - a curve from the manYPoints database [i]

(115−69, 115, 2657)-Net in Base 9 — Upper bound on s

There is no (46, 115, 2658)-net in base 9, because

1 times m-reduction [i] would yield (46, 114, 2658)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 6 085953 600273 008979 412520 380472 350393 567860 326233 726502 459394 663293 600049 543130 042230 269050 974341 997016 541409 > 9¹¹⁴ [i]