MinT - Best Known (77, 114, s)-Nets in Base 8

Best Known (77, 114, s)-Nets in Base 8

(77, 114, 389)-Net over F₈ — Constructive and digital

Digital (77, 114, 389)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (8, 26, 35)-net over F₈, using
  - net from sequence [i] based on digital (8, 34)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₈ with g(F)Â =Â 7, N(F)Â =Â 34, and 1 place with degree 2 [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 (51, 88, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 44, 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]

(77, 114, 576)-Net in Base 8 — Constructive

(77, 114, 576)-net in base 8, using

4 times m-reduction [i] based on (77, 118, 576)-net in base 8, using
- trace code for nets [i] based on (18, 59, 288)-net in base 64, using
  - 4 times m-reduction [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]

(77, 114, 1495)-Net over F₈ — Digital

Digital (77, 114, 1495)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(77, 114, 503988)-Net in Base 8 — Upper bound on s

There is no (77, 114, 503989)-net in base 8, because

1 times m-reduction [i] would yield (77, 113, 503989)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 1 119887 669098 717224 876049 056699 715236 434415 690485 025782 459224 728226 632460 914187 742172 367583 349642 209890 > 8¹¹³ [i]