MinT - Best Known (55, 81, s)-Nets in Base 8

Best Known (55, 81, s)-Nets in Base 8

(55, 81, 371)-Net over F₈ — Constructive and digital

Digital (55, 81, 371)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (2, 15, 17)-net over F₈, using
  - net from sequence [i] based on digital (2, 16)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 2 and N(F) ≥ 17, using
      - a function field by SÃ©mirat [i]
2. digital (40, 66, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 33, 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]

(55, 81, 522)-Net in Base 8 — Constructive

(55, 81, 522)-net in base 8, using

8¹ times duplication [i] based on (54, 80, 522)-net in base 8, using
- base change [i] based on digital (34, 60, 522)-net over F₁₆, using
  - trace code for nets [i] based on digital (4, 30, 261)-net over F₂₅₆, using
    - net from sequence [i] based on digital (4, 260)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

(55, 81, 1239)-Net over F₈ — Digital

Digital (55, 81, 1239)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(55, 81, 342960)-Net in Base 8 — Upper bound on s

There is no (55, 81, 342961)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 14 135303 064862 354075 105978 361421 304174 105646 521947 499972 607090 240242 749904 > 8⁸¹ [i]