MinT - Best Known (70, 70+93, s)-Nets in Base 8

Best Known (70, 70+93, s)-Nets in Base 8

(70, 70+93, 111)-Net over F₈ — Constructive and digital

Digital (70, 163, 111)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (10, 56, 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, 107, 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]

(70, 70+93, 155)-Net over F₈ — Digital

Digital (70, 163, 155)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(70, 70+93, 161)-Net in Base 8

(70, 163, 161)-net in base 8, using

1 times m-reduction [i] based on (70, 164, 161)-net in base 8, using
- base change [i] based on digital (29, 123, 161)-net over F₁₆, using
  - net from sequence [i] based on digital (29, 160)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 29 and N(F) ≥ 161, using
      - a curve from the manYPoints database [i]

(70, 70+93, 3866)-Net in Base 8 — Upper bound on s

There is no (70, 163, 3867)-net in base 8, because

1 times m-reduction [i] would yield (70, 162, 3867)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 200 650715 445620 107367 757251 717188 313054 804789 596665 535714 474691 712052 938589 488782 239345 267980 595593 177484 363782 281474 691415 476463 893692 942061 671200 > 8¹⁶² [i]