MinT - Best Known (72, 169, s)-Nets in Base 8

Best Known (72, 169, s)-Nets in Base 8

(72, 169, 111)-Net over F₈ — Constructive and digital

Digital (72, 169, 111)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (10, 58, 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, 111, 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, 169, 156)-Net over F₈ — Digital

Digital (72, 169, 156)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(72, 169, 161)-Net in Base 8

(72, 169, 161)-net in base 8, using

3 times m-reduction [i] based on (72, 172, 161)-net in base 8, using
- base change [i] based on digital (29, 129, 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]

(72, 169, 3846)-Net in Base 8 — Upper bound on s

There is no (72, 169, 3847)-net in base 8, because

1 times m-reduction [i] would yield (72, 168, 3847)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 52 496296 221341 058866 495775 462292 186684 273489 399023 566710 597733 200612 620025 228839 524494 561497 926946 998921 973422 875256 769743 155994 564740 826596 038843 663984 > 8¹⁶⁸ [i]