MinT - Best Known (259−167, 259, s)-Nets in Base 4

Best Known (259−167, 259, s)-Nets in Base 4

(259−167, 259, 104)-Net over F₄ — Constructive and digital

Digital (92, 259, 104)-net over F₄, using

t-expansion [i] based on digital (73, 259, 104)-net over F₄, using
- net from sequence [i] based on digital (73, 103)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 73 and N(F) ≥ 104, using
    - F₆ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(259−167, 259, 144)-Net over F₄ — Digital

Digital (92, 259, 144)-net over F₄, using

t-expansion [i] based on digital (91, 259, 144)-net over F₄, using
- net from sequence [i] based on digital (91, 143)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 91 and N(F) ≥ 144, using
    - Drinfeld modules of rank 1 [i]

(259−167, 259, 719)-Net in Base 4 — Upper bound on s

There is no (92, 259, 720)-net in base 4, because

1 times m-reduction [i] would yield (92, 258, 720)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 222355 298213 034744 521561 472070 469707 844954 203317 647980 873699 872929 218414 838431 258777 419043 210094 824011 591405 014226 922291 758081 283904 837960 661036 309915 489016 > 4²⁵⁸ [i]