MinT - Best Known (69, 140, s)-Nets in Base 9

Best Known (69, 140, s)-Nets in Base 9

(69, 140, 165)-Net over F₉ — Constructive and digital

Digital (69, 140, 165)-net over F₉, using

t-expansion [i] based on digital (64, 140, 165)-net over F₉, using
- net from sequence [i] based on digital (64, 164)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 64 and N(F) ≥ 165, using
    - T₄ from the second tower of function fields by GarcÃa and Stichtenoth over F₉ [i]

(69, 140, 245)-Net over F₉ — Digital

Digital (69, 140, 245)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(69, 140, 10690)-Net in Base 9 — Upper bound on s

There is no (69, 140, 10691)-net in base 9, because

1 times m-reduction [i] would yield (69, 139, 10691)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 4 364162 640015 817840 756523 911371 147135 792592 697086 282406 694987 227571 947647 219070 735063 521116 344642 217875 481222 094159 055848 245386 728905 > 9¹³⁹ [i]