MinT - Best Known (109, 109+64, s)-Nets in Base 8

Best Known (109, 109+64, s)-Nets in Base 8

(109, 109+64, 354)-Net over F₈ — Constructive and digital

Digital (109, 173, 354)-net over F₈, using

8¹ times duplication [i] based on digital (108, 172, 354)-net over F₈, using
- t-expansion [i] based on digital (93, 172, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 86, 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]

(109, 109+64, 576)-Net in Base 8 — Constructive

(109, 173, 576)-net in base 8, using

8¹ times duplication [i] based on (108, 172, 576)-net in base 8, using
- trace code for nets [i] based on (22, 86, 288)-net in base 64, using
  - 5 times m-reduction [i] based on (22, 91, 288)-net in base 64, using
    - base change [i] based on digital (9, 78, 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]

(109, 109+64, 1060)-Net over F₈ — Digital

Digital (109, 173, 1060)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(109, 109+64, 139331)-Net in Base 8 — Upper bound on s

There is no (109, 173, 139332)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 1 716307 756717 263506 215366 891265 506039 103670 426675 110051 314548 392247 076257 907077 125202 351207 484410 580487 314668 103511 081200 995740 976765 345409 336479 448828 691110 > 8¹⁷³ [i]