MinT - Best Known (127−103, 127, s)-Nets in Base 9

Best Known (127−103, 127, s)-Nets in Base 9

(127−103, 127, 78)-Net over F₉ — Constructive and digital

Digital (24, 127, 78)-net over F₉, using

t-expansion [i] based on digital (22, 127, 78)-net over F₉, using
- net from sequence [i] based on digital (22, 77)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 22 and N(F) ≥ 78, using
    - F₃ from the tower of function fields by GarcÃa and Stichtenoth over F₉ [i]

(127−103, 127, 92)-Net over F₉ — Digital

Digital (24, 127, 92)-net over F₉, using

t-expansion [i] based on digital (23, 127, 92)-net over F₉, using
- net from sequence [i] based on digital (23, 91)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 23 and N(F) ≥ 92, using
    - Drinfeld modules of rank 1 [i]

(127−103, 127, 534)-Net in Base 9 — Upper bound on s

There is no (24, 127, 535)-net in base 9, because

1 times m-reduction [i] would yield (24, 126, 535)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 1 802154 898435 411045 984462 626794 832060 991848 064750 735457 769420 822533 681867 398062 886105 521579 109212 644437 824227 048261 809065 > 9¹²⁶ [i]