MinT - Best Known (73, 170, s)-Nets in Base 8

Best Known (73, 170, s)-Nets in Base 8

(73, 170, 113)-Net over F₈ — Constructive and digital

Digital (73, 170, 113)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (11, 59, 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, 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]

(73, 170, 160)-Net over F₈ — Digital

Digital (73, 170, 160)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(73, 170, 162)-Net in Base 8

(73, 170, 162)-net in base 8, using

2 times m-reduction [i] based on (73, 172, 162)-net in base 8, using
- base change [i] based on digital (30, 129, 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]

(73, 170, 4018)-Net in Base 8 — Upper bound on s

There is no (73, 170, 4019)-net in base 8, because

1 times m-reduction [i] would yield (73, 169, 4019)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 421 671330 379866 930530 131624 728805 122801 751593 064056 267994 956124 098069 797675 733639 443022 462957 047883 343992 889138 284314 840841 019480 110340 666064 821104 316725 > 8¹⁶⁹ [i]