MinT - Best Known (69, 69+103, s)-Nets in Base 8

Best Known (69, 69+103, s)-Nets in Base 8

(69, 69+103, 98)-Net over F₈ — Constructive and digital

Digital (69, 172, 98)-net over F₈, using

t-expansion [i] based on digital (37, 172, 98)-net over F₈, using
- net from sequence [i] based on digital (37, 97)-sequence over F₈, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 37 and N(F) ≥ 98, using
    - F₅ from the tower of function fields over F₈ by van der Geer and van der Vlugt [i]

(69, 69+103, 144)-Net over F₈ — Digital

Digital (69, 172, 144)-net over F₈, using

t-expansion [i] based on digital (45, 172, 144)-net over F₈, using
- net from sequence [i] based on digital (45, 143)-sequence over F₈, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 45 and N(F) ≥ 144, using
    - Drinfeld modules of rank 1 [i]

(69, 69+103, 150)-Net in Base 8

(69, 172, 150)-net in base 8, using

base change [i] based on digital (26, 129, 150)-net over F₁₆, using
- net from sequence [i] based on digital (26, 149)-sequence over F₁₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 26 and N(F) ≥ 150, using
    - a curve from the manYPoints database [i]

(69, 69+103, 2993)-Net in Base 8 — Upper bound on s

There is no (69, 172, 2994)-net in base 8, because

1 times m-reduction [i] would yield (69, 171, 2994)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 27226 969285 278870 938198 404061 715144 477369 331762 053364 541321 798188 663281 399534 980526 979743 213915 798396 434696 225547 525460 351553 608510 136911 258965 333926 086784 > 8¹⁷¹ [i]