MinT - Best Known (152−51, 152, s)-Nets in Base 8

Best Known (152−51, 152, s)-Nets in Base 8

(152−51, 152, 402)-Net over F₈ — Constructive and digital

Digital (101, 152, 402)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (11, 36, 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 (65, 116, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 58, 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]

(152−51, 152, 576)-Net in Base 8 — Constructive

(101, 152, 576)-net in base 8, using

8 times m-reduction [i] based on (101, 160, 576)-net in base 8, using
- trace code for nets [i] based on (21, 80, 288)-net in base 64, using
  - 4 times m-reduction [i] based on (21, 84, 288)-net in base 64, using
    - base change [i] based on digital (9, 72, 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]

(152−51, 152, 1574)-Net over F₈ — Digital

Digital (101, 152, 1574)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(152−51, 152, 414166)-Net in Base 8 — Upper bound on s

There is no (101, 152, 414167)-net in base 8, because

1 times m-reduction [i] would yield (101, 151, 414167)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 23259 523776 519387 070412 361156 157089 568146 098950 440945 930963 868881 469552 327791 663128 779014 696815 187350 465196 576319 270328 272524 217636 336080 > 8¹⁵¹ [i]