MinT - Best Known (158−52, 158, s)-Nets in Base 8

Best Known (158−52, 158, s)-Nets in Base 8

(158−52, 158, 419)-Net over F₈ — Constructive and digital

Digital (106, 158, 419)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (14, 40, 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 (66, 118, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 59, 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]

(158−52, 158, 576)-Net in Base 8 — Constructive

(106, 158, 576)-net in base 8, using

t-expansion [i] based on (105, 158, 576)-net in base 8, using
- 10 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]

(158−52, 158, 1806)-Net over F₈ — Digital

Digital (106, 158, 1806)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(158−52, 158, 463647)-Net in Base 8 — Upper bound on s

There is no (106, 158, 463648)-net in base 8, because

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]