MinT - Best Known (76−23, 76, s)-Nets in Base 8

Best Known (76−23, 76, s)-Nets in Base 8

(76−23, 76, 382)-Net over F₈ — Constructive and digital

Digital (53, 76, 382)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (5, 16, 28)-net over F₈, using
  - net from sequence [i] based on digital (5, 27)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 5 and N(F) ≥ 28, using
      - a function field by SÃ©mirat [i]
2. digital (37, 60, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 30, 177)-net over F₆₄, using
    - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
        a function field by GarcÃa and Quoos [i]

(76−23, 76, 576)-Net in Base 8 — Constructive

(53, 76, 576)-net in base 8, using

trace code for nets [i] based on (15, 38, 288)-net in base 64, using
- 4 times m-reduction [i] based on (15, 42, 288)-net in base 64, using
  - base change [i] based on digital (9, 36, 288)-net over F₁₂₈, using
    - net from sequence [i] based on digital (9, 287)-sequence over F₁₂₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 9 and N(F) ≥ 288, using
        a function field by SÃ©mirat [i]

(76−23, 76, 1715)-Net over F₈ — Digital

Digital (53, 76, 1715)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(76−23, 76, 1007731)-Net in Base 8 — Upper bound on s

There is no (53, 76, 1007732)-net in base 8, because

1 times m-reduction [i] would yield (53, 75, 1007732)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 53 919942 410537 145080 315621 878562 690436 370547 412905 079646 379952 351390 > 8⁷⁵ [i]