MinT - Best Known (165−88, 165, s)-Nets in Base 4

Best Known (165−88, 165, s)-Nets in Base 4

(165−88, 165, 104)-Net over F₄ — Constructive and digital

Digital (77, 165, 104)-net over F₄, using

t-expansion [i] based on digital (73, 165, 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]

(165−88, 165, 112)-Net over F₄ — Digital

Digital (77, 165, 112)-net over F₄, using

t-expansion [i] based on digital (73, 165, 112)-net over F₄, using
- net from sequence [i] based on digital (73, 111)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 73 and N(F) ≥ 112, using
    - Drinfeld modules of rank 1 [i]

(165−88, 165, 1005)-Net in Base 4 — Upper bound on s

There is no (77, 165, 1006)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2235 169384 219096 046940 002153 803007 772242 085045 387234 129372 651956 287083 734167 659338 944436 885119 140607 > 4¹⁶⁵ [i]