MinT - Best Known (123−75, 123, s)-Nets in Base 9

Best Known (123−75, 123, s)-Nets in Base 9

(123−75, 123, 81)-Net over F₉ — Constructive and digital

Digital (48, 123, 81)-net over F₉, using

t-expansion [i] based on digital (32, 123, 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]

(123−75, 123, 82)-Net in Base 9 — Constructive

(48, 123, 82)-net in base 9, using

base change [i] based on digital (7, 82, 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]

(123−75, 123, 163)-Net over F₉ — Digital

Digital (48, 123, 163)-net over F₉, using

net from sequence [i] based on digital (48, 162)-sequence over F₉, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 48 and N(F) ≥ 163, using
  - Drinfeld modules of rank 1 [i]

(123−75, 123, 2543)-Net in Base 9 — Upper bound on s

There is no (48, 123, 2544)-net in base 9, because

1 times m-reduction [i] would yield (48, 122, 2544)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 264 019481 980026 909402 528328 624203 068488 206146 688178 415410 145030 232297 664416 673012 010959 742580 363628 417026 455287 040385 > 9¹²² [i]