MinT - Best Known (63, 63+78, s)-Nets in Base 8

Best Known (63, 63+78, s)-Nets in Base 8

(63, 63+78, 111)-Net over F₈ — Constructive and digital

Digital (63, 141, 111)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (10, 49, 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, 92, 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]

(63, 63+78, 154)-Net over F₈ — Digital

Digital (63, 141, 154)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(63, 63+78, 156)-Net in Base 8

(63, 141, 156)-net in base 8, using

3 times m-reduction [i] based on (63, 144, 156)-net in base 8, using
- base change [i] based on digital (27, 108, 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]

(63, 63+78, 4024)-Net in Base 8 — Upper bound on s

There is no (63, 141, 4025)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 21 777202 349566 003331 396372 060046 427107 941467 826392 515657 050005 131078 976058 376257 703907 287800 010983 512121 524815 511846 042014 218968 > 8¹⁴¹ [i]