MinT - Best Known (53, 53+57, s)-Nets in Base 8

Best Known (53, 53+57, s)-Nets in Base 8

(53, 53+57, 113)-Net over F₈ — Constructive and digital

Digital (53, 110, 113)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (11, 39, 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, 71, 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]

(53, 53+57, 163)-Net over F₈ — Digital

Digital (53, 110, 163)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(53, 53+57, 5273)-Net in Base 8 — Upper bound on s

There is no (53, 110, 5274)-net in base 8, because

1 times m-reduction [i] would yield (53, 109, 5274)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 274 643797 875405 248414 651689 996989 996802 450969 207672 546717 164261 306592 727592 423791 067255 904058 942240 > 8¹⁰⁹ [i]