MinT - Best Known (81, 81+40, s)-Nets in Base 8

Best Known (81, 81+40, s)-Nets in Base 8

(81, 81+40, 388)-Net over F₈ — Constructive and digital

Digital (81, 121, 388)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (7, 27, 34)-net over F₈, using
  - net from sequence [i] based on digital (7, 33)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 7 and N(F) ≥ 34, using
      - a function field by SÃ©mirat [i]
2. digital (54, 94, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 47, 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]

(81, 81+40, 576)-Net in Base 8 — Constructive

(81, 121, 576)-net in base 8, using

5 times m-reduction [i] based on (81, 126, 576)-net in base 8, using
- trace code for nets [i] based on (18, 63, 288)-net in base 64, using
  - base change [i] based on digital (9, 54, 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]

(81, 81+40, 1413)-Net over F₈ — Digital

Digital (81, 121, 1413)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(81, 81+40, 345053)-Net in Base 8 — Upper bound on s

There is no (81, 121, 345054)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 18 788346 493200 267297 479062 007731 445626 920158 618662 615505 101415 367391 557875 271411 421073 448809 192239 010132 458081 > 8¹²¹ [i]