MinT - Best Known (78, 78+37, s)-Nets in Base 8

Best Known (78, 78+37, s)-Nets in Base 8

(78, 78+37, 399)-Net over F₈ — Constructive and digital

Digital (78, 115, 399)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (9, 27, 45)-net over F₈, using
  - net from sequence [i] based on digital (9, 44)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 9 and N(F) ≥ 45, using
      - an explicitly constructive algebraic function field [i]
2. digital (51, 88, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 44, 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]

(78, 78+37, 576)-Net in Base 8 — Constructive

(78, 115, 576)-net in base 8, using

5 times m-reduction [i] based on (78, 120, 576)-net in base 8, using
- trace code for nets [i] based on (18, 60, 288)-net in base 64, using
  - 3 times m-reduction [i] based on (18, 63, 288)-net in base 64, using
    - base change [i] based on digital (9, 54, 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]

(78, 78+37, 1583)-Net over F₈ — Digital

Digital (78, 115, 1583)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(78, 78+37, 565709)-Net in Base 8 — Upper bound on s

There is no (78, 115, 565710)-net in base 8, because

1 times m-reduction [i] would yield (78, 114, 565710)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 8 959135 252758 948961 242830 026593 329962 249358 599936 279638 567703 607888 848103 668739 415638 383317 073503 363496 > 8¹¹⁴ [i]