MinT - Best Known (67, 67+87, s)-Nets in Base 8

Best Known (67, 67+87, s)-Nets in Base 8

(67, 67+87, 111)-Net over F₈ — Constructive and digital

Digital (67, 154, 111)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (10, 53, 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, 101, 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]

(67, 67+87, 153)-Net over F₈ — Digital

Digital (67, 154, 153)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(67, 67+87, 156)-Net in Base 8

(67, 154, 156)-net in base 8, using

6 times m-reduction [i] based on (67, 160, 156)-net in base 8, using
- base change [i] based on digital (27, 120, 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]

(67, 67+87, 3914)-Net in Base 8 — Upper bound on s

There is no (67, 154, 3915)-net in base 8, because

1 times m-reduction [i] would yield (67, 153, 3915)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 1 493796 882023 965481 537916 844048 533923 527923 068897 718588 922240 152822 469070 343027 889522 633234 319505 702303 722514 352202 749596 949574 047058 826320 > 8¹⁵³ [i]