MinT - Best Known (81−28, 81, s)-Nets in Base 8

Best Known (81−28, 81, s)-Nets in Base 8

(81−28, 81, 354)-Net over F₈ — Constructive and digital

Digital (53, 81, 354)-net over F₈, using

11 times m-reduction [i] based on digital (53, 92, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 46, 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]

(81−28, 81, 518)-Net in Base 8 — Constructive

(53, 81, 518)-net in base 8, using

8¹ times duplication [i] based on (52, 80, 518)-net in base 8, using
- base change [i] based on digital (32, 60, 518)-net over F₁₆, using
  - trace code for nets [i] based on digital (2, 30, 259)-net over F₂₅₆, using
    - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

(81−28, 81, 812)-Net over F₈ — Digital

Digital (53, 81, 812)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(81−28, 81, 144995)-Net in Base 8 — Upper bound on s

There is no (53, 81, 144996)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 14 135296 522499 626097 261166 937971 983364 656121 167357 787148 924977 633995 636856 > 8⁸¹ [i]