MinT - Best Known (68, 163, s)-Nets in Base 8

Best Known (68, 163, s)-Nets in Base 8

(68, 163, 99)-Net over F₈ — Constructive and digital

Digital (68, 163, 99)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (7, 54, 34)-net over F₈, using
  - net from sequence [i] based on digital (7, 33)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 7 and N(F) ≥ 34, using
      - a function field by SÃ©mirat [i]
2. digital (14, 109, 65)-net over F₈, using
  - net from sequence [i] based on digital (14, 64)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 14 and N(F) ≥ 65, using
      - the Suzuki function field over F₈ [i]

(68, 163, 144)-Net over F₈ — Digital

Digital (68, 163, 144)-net over F₈, using

t-expansion [i] based on digital (45, 163, 144)-net over F₈, using
- net from sequence [i] based on digital (45, 143)-sequence over F₈, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 45 and N(F) ≥ 144, using
    - Drinfeld modules of rank 1 [i]

(68, 163, 156)-Net in Base 8

(68, 163, 156)-net in base 8, using

1 times m-reduction [i] based on (68, 164, 156)-net in base 8, using
- base change [i] based on digital (27, 123, 156)-net over F₁₆, using
  - net from sequence [i] based on digital (27, 155)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 27 and N(F) ≥ 156, using
      - a curve from the manYPoints database [i]

(68, 163, 3372)-Net in Base 8 — Upper bound on s

There is no (68, 163, 3373)-net in base 8, because

1 times m-reduction [i] would yield (68, 162, 3373)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 200 570245 403889 214342 833439 073312 486125 845410 775284 589957 877050 954788 015681 354538 787676 371437 286884 131354 352056 627821 193010 684939 676021 448460 087552 > 8¹⁶² [i]