MinT - Best Known (51, 104, s)-Nets in Base 8

Best Known (51, 104, s)-Nets in Base 8

(51, 104, 113)-Net over F₈ — Constructive and digital

Digital (51, 104, 113)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (11, 37, 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, 67, 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]

(51, 104, 166)-Net over F₈ — Digital

Digital (51, 104, 166)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(51, 104, 5683)-Net in Base 8 — Upper bound on s

There is no (51, 104, 5684)-net in base 8, because

1 times m-reduction [i] would yield (51, 103, 5684)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 1047 303538 743214 607761 900138 831519 783433 296366 812015 933929 588822 438287 959519 910890 620259 518468 > 8¹⁰³ [i]