MinT - Best Known (97, 97+51, s)-Nets in Base 8

Best Known (97, 97+51, s)-Nets in Base 8

(97, 97+51, 388)-Net over F₈ — Constructive and digital

Digital (97, 148, 388)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (7, 32, 34)-net over F₈, using
  - net from sequence [i] based on digital (7, 33)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 7 and N(F) ≥ 34, 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]

(97, 97+51, 576)-Net in Base 8 — Constructive

(97, 148, 576)-net in base 8, using

6 times m-reduction [i] based on (97, 154, 576)-net in base 8, using
- trace code for nets [i] based on (20, 77, 288)-net in base 64, using
  - base change [i] based on digital (9, 66, 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]

(97, 97+51, 1336)-Net over F₈ — Digital

Digital (97, 148, 1336)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(97, 97+51, 296943)-Net in Base 8 — Upper bound on s

There is no (97, 148, 296944)-net in base 8, because

1 times m-reduction [i] would yield (97, 147, 296944)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 5 678563 759469 093432 965848 192383 895914 357842 648291 485429 204126 639289 869713 860111 254542 079659 852704 974936 994606 533969 629020 413739 185842 > 8¹⁴⁷ [i]