MinT - Best Known (59, 59+137, s)-Nets in Base 4

Best Known (59, 59+137, s)-Nets in Base 4

(59, 59+137, 66)-Net over F₄ — Constructive and digital

Digital (59, 196, 66)-net over F₄, using

t-expansion [i] based on digital (49, 196, 66)-net over F₄, using
- net from sequence [i] based on digital (49, 65)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 49 and N(F) ≥ 66, using
    - T₆ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(59, 59+137, 91)-Net over F₄ — Digital

Digital (59, 196, 91)-net over F₄, using

t-expansion [i] based on digital (50, 196, 91)-net over F₄, using
- net from sequence [i] based on digital (50, 90)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 50 and N(F) ≥ 91, using
    - a curve from the manYPoints database [i]

(59, 59+137, 410)-Net in Base 4 — Upper bound on s

There is no (59, 196, 411)-net in base 4, because

1 times m-reduction [i] would yield (59, 195, 411)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2626 004723 029120 712222 952871 412069 719506 712928 596980 277121 347089 887080 656674 486546 686842 409507 535228 440146 392216 851420 > 4¹⁹⁵ [i]