MinT - Best Known (125−67, 125, s)-Nets in Base 8

Best Known (125−67, 125, s)-Nets in Base 8

(125−67, 125, 113)-Net over F₈ — Constructive and digital

Digital (58, 125, 113)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (11, 44, 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, 81, 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]

(125−67, 125, 158)-Net over F₈ — Digital

Digital (58, 125, 158)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(125−67, 125, 4631)-Net in Base 8 — Upper bound on s

There is no (58, 125, 4632)-net in base 8, because

1 times m-reduction [i] would yield (58, 124, 4632)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 9633 262373 916349 337781 559395 423208 164893 820600 375617 926629 571758 712269 425137 640641 424064 799451 639128 778349 100240 > 8¹²⁴ [i]