MinT - Best Known (50, 50+51, s)-Nets in Base 8

Best Known (50, 50+51, s)-Nets in Base 8

(50, 50+51, 113)-Net over F₈ — Constructive and digital

Digital (50, 101, 113)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (11, 36, 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, 65, 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]

(50, 50+51, 169)-Net over F₈ — Digital

Digital (50, 101, 169)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(50, 50+51, 5939)-Net in Base 8 — Upper bound on s

There is no (50, 101, 5940)-net in base 8, because

1 times m-reduction [i] would yield (50, 100, 5940)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 2 041119 059350 253624 581195 272765 242626 894601 170220 425750 822499 402888 346028 738037 585778 827932 > 8¹⁰⁰ [i]