MinT - Best Known (75−39, 75, s)-Nets in Base 8

Best Known (75−39, 75, s)-Nets in Base 8

(75−39, 75, 89)-Net over F₈ — Constructive and digital

Digital (36, 75, 89)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (3, 22, 24)-net over F₈, using
  - net from sequence [i] based on digital (3, 23)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 3 and N(F) ≥ 24, using
      - the Klein quartic over F₈ [i]
2. digital (14, 53, 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]

(75−39, 75, 119)-Net over F₈ — Digital

Digital (36, 75, 119)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(75−39, 75, 3715)-Net in Base 8 — Upper bound on s

There is no (36, 75, 3716)-net in base 8, because

1 times m-reduction [i] would yield (36, 74, 3716)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 6 747648 144409 760480 939066 362441 368516 824824 225130 773503 323565 793320 > 8⁷⁴ [i]