MinT - Best Known (59, 59+135, s)-Nets in Base 4

Best Known (59, 59+135, s)-Nets in Base 4

(59, 59+135, 66)-Net over F₄ — Constructive and digital

Digital (59, 194, 66)-net over F₄, using

t-expansion [i] based on digital (49, 194, 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+135, 91)-Net over F₄ — Digital

Digital (59, 194, 91)-net over F₄, using

t-expansion [i] based on digital (50, 194, 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+135, 413)-Net in Base 4 — Upper bound on s

There is no (59, 194, 414)-net in base 4, because

1 times m-reduction [i] would yield (59, 193, 414)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 176 944750 802249 488966 942503 538263 012224 142366 426769 106352 027329 996181 885689 226971 122040 212845 555162 457678 269138 386500 > 4¹⁹³ [i]