MinT - Best Known (159−53, 159, s)-Nets in Base 8

Best Known (159−53, 159, s)-Nets in Base 8

(159−53, 159, 402)-Net over F₈ — Constructive and digital

Digital (106, 159, 402)-net over F₈, using

1 times m-reduction [i] based on digital (106, 160, 402)-net over F₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (11, 38, 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 (68, 122, 354)-net over F₈, using
    - trace code for nets [i] based on digital (7, 61, 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]

(159−53, 159, 576)-Net in Base 8 — Constructive

(106, 159, 576)-net in base 8, using

t-expansion [i] based on (105, 159, 576)-net in base 8, using
- 9 times m-reduction [i] based on (105, 168, 576)-net in base 8, using
  - trace code for nets [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]

(159−53, 159, 1694)-Net over F₈ — Digital

Digital (106, 159, 1694)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(159−53, 159, 463647)-Net in Base 8 — Upper bound on s

There is no (106, 159, 463648)-net in base 8, because

1 times m-reduction [i] would yield (106, 158, 463648)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 48777 893763 664512 309194 385117 388789 233930 620107 335967 671590 298936 932689 957936 130659 493227 236776 121253 771740 214654 520184 181917 923684 307345 905351 > 8¹⁵⁸ [i]