MinT - Best Known (64, 64+81, s)-Nets in Base 8

Best Known (64, 64+81, s)-Nets in Base 8

(64, 64+81, 111)-Net over F₈ — Constructive and digital

Digital (64, 145, 111)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (10, 50, 46)-net over F₈, using
  - net from sequence [i] based on digital (10, 45)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₈ with g(F)Â =Â 9, N(F)Â =Â 45, and 1 place with degree 2 [i] based on function field F/F₈ with g(F) = 9 and N(F) ≥ 45, using an explicitly constructive algebraic function field [i]
2. digital (14, 95, 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]

(64, 64+81, 152)-Net over F₈ — Digital

Digital (64, 145, 152)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(64, 64+81, 156)-Net in Base 8

(64, 145, 156)-net in base 8, using

3 times m-reduction [i] based on (64, 148, 156)-net in base 8, using
- base change [i] based on digital (27, 111, 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]

(64, 64+81, 3991)-Net in Base 8 — Upper bound on s

There is no (64, 145, 3992)-net in base 8, because

1 times m-reduction [i] would yield (64, 144, 3992)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 11196 137338 721177 088047 318261 734112 244886 968208 656601 233296 702746 419214 399218 732776 112243 373532 980584 990366 915461 637788 221446 033000 > 8¹⁴⁴ [i]