MinT - Best Known (168−112, 168, s)-Nets in Base 4

Best Known (168−112, 168, s)-Nets in Base 4

(168−112, 168, 66)-Net over F₄ — Constructive and digital

Digital (56, 168, 66)-net over F₄, using

t-expansion [i] based on digital (49, 168, 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]

(168−112, 168, 91)-Net over F₄ — Digital

Digital (56, 168, 91)-net over F₄, using

t-expansion [i] based on digital (50, 168, 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]

(168−112, 168, 418)-Net in Base 4 — Upper bound on s

There is no (56, 168, 419)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 146375 822222 993131 210634 987673 716196 983878 328580 960322 703165 872275 934528 334250 870207 358310 196274 206160 > 4¹⁶⁸ [i]