MinT - Best Known (49, 49+77, s)-Nets in Base 9

Best Known (49, 49+77, s)-Nets in Base 9

(49, 49+77, 81)-Net over F₉ — Constructive and digital

Digital (49, 126, 81)-net over F₉, using

t-expansion [i] based on digital (32, 126, 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]

(49, 49+77, 82)-Net in Base 9 — Constructive

(49, 126, 82)-net in base 9, using

base change [i] based on digital (7, 84, 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]

(49, 49+77, 168)-Net over F₉ — Digital

Digital (49, 126, 168)-net over F₉, using

net from sequence [i] based on digital (49, 167)-sequence over F₉, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 49 and N(F) ≥ 168, using
  - Drinfeld modules of rank 1 [i]

(49, 49+77, 2563)-Net in Base 9 — Upper bound on s

There is no (49, 126, 2564)-net in base 9, because

1 times m-reduction [i] would yield (49, 125, 2564)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 193360 381406 050436 576042 371321 346107 974273 567318 245283 871827 589064 242562 668762 865459 863074 824471 179361 454373 319011 463745 > 9¹²⁵ [i]