MinT - Best Known (89−31, 89, s)-Nets in Base 8

Best Known (89−31, 89, s)-Nets in Base 8

(89−31, 89, 354)-Net over F₈ — Constructive and digital

Digital (58, 89, 354)-net over F₈, using

13 times m-reduction [i] based on digital (58, 102, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 51, 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]

(89−31, 89, 518)-Net in Base 8 — Constructive

(58, 89, 518)-net in base 8, using

8¹ times duplication [i] based on (57, 88, 518)-net in base 8, using
- base change [i] based on digital (35, 66, 518)-net over F₁₆, using
  - trace code for nets [i] based on digital (2, 33, 259)-net over F₂₅₆, using
    - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

(89−31, 89, 837)-Net over F₈ — Digital

Digital (58, 89, 837)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(89−31, 89, 182294)-Net in Base 8 — Upper bound on s

There is no (58, 89, 182295)-net in base 8, because

1 times m-reduction [i] would yield (58, 88, 182295)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 29 642788 501712 430862 452615 841040 261917 610417 345440 513791 283253 519827 755502 032000 > 8⁸⁸ [i]