MinT - Best Known (86, 86+48, s)-Nets in Base 8

Best Known (86, 86+48, s)-Nets in Base 8

(86, 86+48, 363)-Net over F₈ — Constructive and digital

Digital (86, 134, 363)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (0, 24, 9)-net over F₈, using
  - net from sequence [i] based on digital (0, 8)-sequence over F₈, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 0 and N(F) ≥ 9, using
      - the rational function field F₈(x) [i]
    - Niederreiter sequence [i]
2. digital (62, 110, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 55, 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]

(86, 86+48, 576)-Net in Base 8 — Constructive

(86, 134, 576)-net in base 8, using

trace code for nets [i] based on (19, 67, 288)-net in base 64, using
- 3 times m-reduction [i] based on (19, 70, 288)-net in base 64, using
  - base change [i] based on digital (9, 60, 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]

(86, 86+48, 1009)-Net over F₈ — Digital

Digital (86, 134, 1009)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(86, 86+48, 154338)-Net in Base 8 — Upper bound on s

There is no (86, 134, 154339)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 10 329320 663855 363563 878319 473918 522705 468904 623460 756231 645466 817479 935446 366567 491388 067880 563241 336425 405426 939846 228132 > 8¹³⁴ [i]