MinT - Best Known (120−71, 120, s)-Nets in Base 9

Best Known (120−71, 120, s)-Nets in Base 9

(120−71, 120, 81)-Net over F₉ — Constructive and digital

Digital (49, 120, 81)-net over F₉, using

t-expansion [i] based on digital (32, 120, 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]

(120−71, 120, 88)-Net in Base 9 — Constructive

(49, 120, 88)-net in base 9, using

base change [i] based on digital (9, 80, 88)-net over F₂₇, using
- net from sequence [i] based on digital (9, 87)-sequence over F₂₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 9 and N(F) ≥ 88, using
    - a function field by SÃ©mirat [i]

(120−71, 120, 168)-Net over F₉ — Digital

Digital (49, 120, 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]

(120−71, 120, 3030)-Net in Base 9 — Upper bound on s

There is no (49, 120, 3031)-net in base 9, because

1 times m-reduction [i] would yield (49, 119, 3031)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 359769 105679 221027 716387 628401 567947 477420 132733 396957 269754 664975 431273 167078 178471 576318 898754 965105 451361 903145 > 9¹¹⁹ [i]