MinT - Best Known (115−34, 115, s)-Nets in Base 8

Best Known (115−34, 115, s)-Nets in Base 8

(115−34, 115, 419)-Net over F₈ — Constructive and digital

Digital (81, 115, 419)-net over F₈, using

t-expansion [i] based on digital (80, 115, 419)-net over F₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (14, 31, 65)-net over F₈, using
    - net from sequence [i] based on digital (14, 64)-sequence over F₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 14 and N(F) ≥ 65, using
        the Suzuki function field over F₈ [i]
  2. digital (49, 84, 354)-net over F₈, using
    - trace code for nets [i] based on digital (7, 42, 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]

(115−34, 115, 576)-Net in Base 8 — Constructive

(81, 115, 576)-net in base 8, using

11 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]

(115−34, 115, 2655)-Net over F₈ — Digital

Digital (81, 115, 2655)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(115−34, 115, 1318201)-Net in Base 8 — Upper bound on s

There is no (81, 115, 1318202)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 71 672078 462363 736515 812412 442816 684306 091285 158143 491692 687782 811840 129282 767666 132656 825297 471978 634767 > 8¹¹⁵ [i]