MinT - Best Known (78, 78+42, s)-Nets in Base 8

Best Known (78, 78+42, s)-Nets in Base 8

(78, 78+42, 368)-Net over F₈ — Constructive and digital

Digital (78, 120, 368)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (1, 22, 14)-net over F₈, using
  - net from sequence [i] based on digital (1, 13)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 1 and N(F) ≥ 14, using
      - F₂ from the tower of function fields over F₈ by van der Geer and van der Vlugt [i]
2. digital (56, 98, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 49, 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+42, 576)-Net in Base 8 — Constructive

(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+42, 1034)-Net over F₈ — Digital

Digital (78, 120, 1034)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(78, 78+42, 179422)-Net in Base 8 — Upper bound on s

There is no (78, 120, 179423)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 2 348637 983907 195629 255721 663250 203403 687344 707044 109928 863166 808641 651828 817798 223917 643190 383570 708711 850082 > 8¹²⁰ [i]