MinT - Best Known (139−91, 139, s)-Nets in Base 9

Best Known (139−91, 139, s)-Nets in Base 9

(139−91, 139, 81)-Net over F₉ — Constructive and digital

Digital (48, 139, 81)-net over F₉, using

t-expansion [i] based on digital (32, 139, 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]

(139−91, 139, 163)-Net over F₉ — Digital

Digital (48, 139, 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]

(139−91, 139, 1832)-Net in Base 9 — Upper bound on s

There is no (48, 139, 1833)-net in base 9, because

1 times m-reduction [i] would yield (48, 138, 1833)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 495029 689529 388346 557713 321079 949595 965369 374552 435262 857290 983147 926975 826396 194674 260752 051855 806680 027710 924641 425611 238414 784297 > 9¹³⁸ [i]