MinT - Best Known (86−72, 86, s)-Nets in Base 64

Best Known (86−72, 86, s)-Nets in Base 64

(86−72, 86, 177)-Net over F₆₄ — Constructive and digital

Digital (14, 86, 177)-net over F₆₄, using

t-expansion [i] based on digital (7, 86, 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]

(86−72, 86, 257)-Net over F₆₄ — Digital

Digital (14, 86, 257)-net over F₆₄, using

t-expansion [i] based on digital (12, 86, 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]

(86−72, 86, 4661)-Net in Base 64 — Upper bound on s

There is no (14, 86, 4662)-net in base 64, because

the generalized Rao bound for nets shows that 64^mÂ â‰¥ 216156 457329 073960 152508 403496 449188 383371 629045 352625 386766 062477 910065 069372 781870 370370 342005 062555 052626 142271 542650 558943 023654 381223 958296 596666 577567 > 64⁸⁶ [i]