MinT - Best Known (86, 86+62, s)-Nets in Base 8

Best Known (86, 86+62, s)-Nets in Base 8

(86, 86+62, 354)-Net over F₈ — Constructive and digital

Digital (86, 148, 354)-net over F₈, using

10 times m-reduction [i] based on digital (86, 158, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 79, 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]

(86, 86+62, 514)-Net over F₈ — Digital

Digital (86, 148, 514)-net over F₈, using

trace code for nets [i] based on digital (12, 74, 257)-net over F₆₄, using
- net from sequence [i] based on digital (12, 256)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 12 and N(F) ≥ 257, using
    - K_1,2 from the tower of function fields by Niederreiter and Xing based on the tower by GarcÃa and Stichtenoth over F₆₄ [i]

(86, 86+62, 36326)-Net in Base 8 — Upper bound on s

There is no (86, 148, 36327)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 45 428702 085119 562566 811655 932094 067975 793262 504459 145101 923154 359565 389005 654863 225302 668075 212713 074052 183569 098583 334282 814146 048000 > 8¹⁴⁸ [i]