MinT - Best Known (106−36, 106, s)-Nets in Base 8

Best Known (106−36, 106, s)-Nets in Base 8

(106−36, 106, 371)-Net over F₈ — Constructive and digital

Digital (70, 106, 371)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (2, 20, 17)-net over F₈, using
  - net from sequence [i] based on digital (2, 16)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 2 and N(F) ≥ 17, using
      - a function field by SÃ©mirat [i]
2. digital (50, 86, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 43, 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]

(106−36, 106, 576)-Net in Base 8 — Constructive

(70, 106, 576)-net in base 8, using

trace code for nets [i] based on (17, 53, 288)-net in base 64, using
- 3 times m-reduction [i] based on (17, 56, 288)-net in base 64, using
  - base change [i] based on digital (9, 48, 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]

(106−36, 106, 1097)-Net over F₈ — Digital

Digital (70, 106, 1097)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(106−36, 106, 224495)-Net in Base 8 — Upper bound on s

There is no (70, 106, 224496)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 534028 137248 657417 968649 481417 662909 663966 213193 957409 712396 653269 863605 343147 083567 005880 564262 > 8¹⁰⁶ [i]