MinT - Best Known (69, 160, s)-Nets in Base 8

Best Known (69, 160, s)-Nets in Base 8

(69, 160, 111)-Net over F₈ — Constructive and digital

Digital (69, 160, 111)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (10, 55, 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, 105, 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]

(69, 160, 154)-Net over F₈ — Digital

Digital (69, 160, 154)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(69, 160, 161)-Net in Base 8

(69, 160, 161)-net in base 8, using

base change [i] based on digital (29, 120, 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]

(69, 160, 3879)-Net in Base 8 — Upper bound on s

There is no (69, 160, 3880)-net in base 8, because

1 times m-reduction [i] would yield (69, 159, 3880)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 390287 188294 524018 806382 131486 130079 608533 517167 286410 254755 558505 902896 589377 983832 651715 264847 159552 436720 164527 150700 286760 947773 557641 085088 > 8¹⁵⁹ [i]