MinT - Best Known (72, 72+99, s)-Nets in Base 8

Best Known (72, 72+99, s)-Nets in Base 8

(72, 72+99, 110)-Net over F₈ — Constructive and digital

Digital (72, 171, 110)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (9, 58, 45)-net over F₈, using
  - net from sequence [i] based on digital (9, 44)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [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, 113, 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, 72+99, 152)-Net over F₈ — Digital

Digital (72, 171, 152)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(72, 72+99, 161)-Net in Base 8

(72, 171, 161)-net in base 8, using

1 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, 72+99, 3679)-Net in Base 8 — Upper bound on s

There is no (72, 171, 3680)-net in base 8, because

1 times m-reduction [i] would yield (72, 170, 3680)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 3390 831641 639989 680934 073371 453183 856204 120838 214395 756451 669115 771073 047664 581760 835848 208868 335835 719804 994072 675456 854097 036385 625922 894785 569676 109022 > 8¹⁷⁰ [i]