MinT - Best Known (68, 97, s)-Nets in Base 8

Best Known (68, 97, s)-Nets in Base 8

(68, 97, 402)-Net over F₈ — Constructive and digital

Digital (68, 97, 402)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (11, 25, 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 (43, 72, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 36, 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]

(68, 97, 576)-Net in Base 8 — Constructive

(68, 97, 576)-net in base 8, using

5 times m-reduction [i] based on (68, 102, 576)-net in base 8, using
- trace code for nets [i] based on (17, 51, 288)-net in base 64, using
  - 5 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]

(68, 97, 2184)-Net over F₈ — Digital

Digital (68, 97, 2184)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(68, 97, 1345779)-Net in Base 8 — Upper bound on s

There is no (68, 97, 1345780)-net in base 8, because

1 times m-reduction [i] would yield (68, 96, 1345780)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 497 324194 725560 624935 352536 491766 459987 372124 769316 115372 224317 676275 622564 675252 026404 > 8⁹⁶ [i]