MinT - Best Known (89, 89+49, s)-Nets in Base 8

Best Known (89, 89+49, s)-Nets in Base 8

(89, 89+49, 371)-Net over F₈ — Constructive and digital

Digital (89, 138, 371)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (2, 26, 17)-net over F₈, using
  - net from sequence [i] based on digital (2, 16)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 2 and N(F) ≥ 17, using
      - a function field by SÃ©mirat [i]
2. digital (63, 112, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 56, 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]

(89, 89+49, 576)-Net in Base 8 — Constructive

(89, 138, 576)-net in base 8, using

2 times m-reduction [i] based on (89, 140, 576)-net in base 8, using
- trace code for nets [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]

(89, 89+49, 1081)-Net over F₈ — Digital

Digital (89, 138, 1081)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(89, 89+49, 200157)-Net in Base 8 — Upper bound on s

There is no (89, 138, 200158)-net in base 8, because

1 times m-reduction [i] would yield (89, 137, 200158)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 5289 023603 278799 733143 590430 041225 004971 682632 625220 931819 659054 997546 268867 742156 495347 162075 052601 806278 911219 111142 903623 > 8¹³⁷ [i]