MinT - Best Known (72, 167, s)-Nets in Base 8

Best Known (72, 167, s)-Nets in Base 8

(72, 167, 113)-Net over F₈ — Constructive and digital

Digital (72, 167, 113)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (11, 58, 48)-net over F₈, using
  - net from sequence [i] based on digital (11, 47)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 11 and N(F) ≥ 48, using
      - a function field by SÃ©mirat [i]
2. digital (14, 109, 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]

(72, 167, 160)-Net over F₈ — Digital

Digital (72, 167, 160)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(72, 167, 162)-Net in Base 8

(72, 167, 162)-net in base 8, using

1 times m-reduction [i] based on (72, 168, 162)-net in base 8, using
- base change [i] based on digital (30, 126, 162)-net over F₁₆, using
  - net from sequence [i] based on digital (30, 161)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 30 and N(F) ≥ 162, using
      - a curve from the manYPoints database [i]

(72, 167, 4031)-Net in Base 8 — Upper bound on s

There is no (72, 167, 4032)-net in base 8, because

1 times m-reduction [i] would yield (72, 166, 4032)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 824042 421087 037512 950701 774031 239729 457770 554175 466741 477242 798973 769678 550074 077713 753555 467047 096735 246342 355121 695071 246926 698451 386529 152764 534215 > 8¹⁶⁶ [i]