MinT - Best Known (86−45, 86, s)-Nets in Base 64

Best Known (86−45, 86, s)-Nets in Base 64

(86−45, 86, 513)-Net over F₆₄ — Constructive and digital

Digital (41, 86, 513)-net over F₆₄, using

t-expansion [i] based on digital (28, 86, 513)-net over F₆₄, using
- net from sequence [i] based on digital (28, 512)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 28 and N(F) ≥ 513, using
    - the Hermitian function field over F₆₄ [i]

(86−45, 86, 951)-Net over F₆₄ — Digital

Digital (41, 86, 951)-net over F₆₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(86−45, 86, 1367513)-Net in Base 64 — Upper bound on s

There is no (41, 86, 1367514)-net in base 64, because

1 times m-reduction [i] would yield (41, 85, 1367514)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 3351 972762 202905 488483 139044 061240 551369 086415 366672 082065 533862 894318 785661 705452 983278 831849 601909 257048 355432 406514 562176 421498 481782 941125 073106 789580 > 64⁸⁵ [i]