MinT - Best Known (138−103, 138, s)-Nets in Base 9

Best Known (138−103, 138, s)-Nets in Base 9

(138−103, 138, 81)-Net over F₉ — Constructive and digital

Digital (35, 138, 81)-net over F₉, using

t-expansion [i] based on digital (32, 138, 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]

(138−103, 138, 128)-Net over F₉ — Digital

Digital (35, 138, 128)-net over F₉, using

t-expansion [i] based on digital (33, 138, 128)-net over F₉, using
- net from sequence [i] based on digital (33, 127)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 33 and N(F) ≥ 128, using
    - a curve from the manYPoints database [i]

(138−103, 138, 877)-Net in Base 9 — Upper bound on s

There is no (35, 138, 878)-net in base 9, because

1 times m-reduction [i] would yield (35, 137, 878)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 56916 055618 817975 906299 736102 141882 985265 206438 801763 288091 443036 749016 250354 255665 682335 904551 173462 368889 606808 398368 616612 702225 > 9¹³⁷ [i]