MinT - Best Known (80, 80+37, s)-Nets in Base 8

Best Known (80, 80+37, s)-Nets in Base 8

(80, 80+37, 402)-Net over F₈ — Constructive and digital

Digital (80, 117, 402)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (11, 29, 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 (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]

(80, 80+37, 576)-Net in Base 8 — Constructive

(80, 117, 576)-net in base 8, using

7 times m-reduction [i] based on (80, 124, 576)-net in base 8, using
- trace code for nets [i] based on (18, 62, 288)-net in base 64, using
  - 1 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]

(80, 80+37, 1775)-Net over F₈ — Digital

Digital (80, 117, 1775)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(80, 80+37, 712752)-Net in Base 8 — Upper bound on s

There is no (80, 117, 712753)-net in base 8, because

1 times m-reduction [i] would yield (80, 116, 712753)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 573 387337 880722 268291 417891 913794 797881 565300 975068 462977 236517 848205 593904 185950 004338 515967 453190 188112 > 8¹¹⁶ [i]