MinT - Best Known (172−114, 172, s)-Nets in Base 8

Best Known (172−114, 172, s)-Nets in Base 8

(172−114, 172, 98)-Net over F₈ — Constructive and digital

Digital (58, 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]

(172−114, 172, 144)-Net over F₈ — Digital

Digital (58, 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]

(172−114, 172, 1639)-Net in Base 8 — Upper bound on s

There is no (58, 172, 1640)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 221368 578429 400071 450514 154579 438717 504552 259752 148913 964551 393092 505186 196959 027697 556911 481667 214131 479038 012785 741557 770967 016687 137408 632434 012333 571768 > 8¹⁷² [i]