MinT - Best Known (68, 68+61, s)-Nets in Base 9

Best Known (68, 68+61, s)-Nets in Base 9

(68, 68+61, 232)-Net over F₉ — Constructive and digital

Digital (68, 129, 232)-net over F₉, using

3 times m-reduction [i] based on digital (68, 132, 232)-net over F₉, using
- trace code for nets [i] based on digital (2, 66, 116)-net over F₈₁, using
  - net from sequence [i] based on digital (2, 115)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 2 and N(F) ≥ 116, using
      - a function field by SÃ©mirat [i]

(68, 68+61, 310)-Net over F₉ — Digital

Digital (68, 129, 310)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(68, 68+61, 17729)-Net in Base 9 — Upper bound on s

There is no (68, 129, 17730)-net in base 9, because

1 times m-reduction [i] would yield (68, 128, 17730)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 139 219984 940956 715417 520934 772260 898533 064956 854618 576594 214476 249245 943063 358859 055639 964947 068517 506463 252192 722095 799329 > 9¹²⁸ [i]