MinT - Best Known (135−44, 135, s)-Nets in Base 8

Best Known (135−44, 135, s)-Nets in Base 8

(135−44, 135, 402)-Net over F₈ — Constructive and digital

Digital (91, 135, 402)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (11, 33, 48)-net over F₈, using
  - net from sequence [i] based on digital (11, 47)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 11 and N(F) ≥ 48, using
      - a function field by SÃ©mirat [i]
2. digital (58, 102, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 51, 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]

(135−44, 135, 576)-Net in Base 8 — Constructive

(91, 135, 576)-net in base 8, using

t-expansion [i] based on (89, 135, 576)-net in base 8, using
- 5 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]

(135−44, 135, 1672)-Net over F₈ — Digital

Digital (91, 135, 1672)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(135−44, 135, 450225)-Net in Base 8 — Upper bound on s

There is no (91, 135, 450226)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 82 632781 708798 153231 559579 871439 800297 976584 588895 490214 283665 826527 031341 716009 607485 632724 021271 617222 548355 648562 976144 > 8¹³⁵ [i]