MinT - Best Known (36, 36+49, s)-Nets in Base 9

Best Known (36, 36+49, s)-Nets in Base 9

(36, 36+49, 81)-Net over F₉ — Constructive and digital

Digital (36, 85, 81)-net over F₉, using

t-expansion [i] based on digital (32, 85, 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]

(36, 36+49, 82)-Net in Base 9 — Constructive

(36, 85, 82)-net in base 9, using

2 times m-reduction [i] based on (36, 87, 82)-net in base 9, using
- base change [i] based on digital (7, 58, 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]

(36, 36+49, 128)-Net over F₉ — Digital

Digital (36, 85, 128)-net over F₉, using

t-expansion [i] based on digital (33, 85, 128)-net over F₉, using
- net from sequence [i] based on digital (33, 127)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 33 and N(F) ≥ 128, using
    - a curve from the manYPoints database [i]

(36, 36+49, 2665)-Net in Base 9 — Upper bound on s

There is no (36, 85, 2666)-net in base 9, because

1 times m-reduction [i] would yield (36, 84, 2666)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 144 059740 057676 233027 792273 861112 266181 204817 828108 257441 512192 137088 440920 732289 > 9⁸⁴ [i]