MinT - Best Known (42, 89, s)-Nets in Base 64

Best Known (42, 89, s)-Nets in Base 64

(42, 89, 513)-Net over F₆₄ — Constructive and digital

Digital (42, 89, 513)-net over F₆₄, using

t-expansion [i] based on digital (28, 89, 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]

(42, 89, 915)-Net over F₆₄ — Digital

Digital (42, 89, 915)-net over F₆₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(42, 89, 1218185)-Net in Base 64 — Upper bound on s

There is no (42, 89, 1218186)-net in base 64, because

1 times m-reduction [i] would yield (42, 88, 1218186)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 878 707417 069271 648664 525689 225546 945411 959667 118395 175866 823653 358825 107326 089358 355651 155253 942572 166458 578506 007842 856013 806177 504842 719378 105246 919306 116384 > 64⁸⁸ [i]