MinT - Best Known (165−77, 165, s)-Nets in Base 4

Best Known (165−77, 165, s)-Nets in Base 4

(165−77, 165, 104)-Net over F₄ — Constructive and digital

Digital (88, 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]

(165−77, 165, 151)-Net over F₄ — Digital

Digital (88, 165, 151)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(165−77, 165, 1955)-Net in Base 4 — Upper bound on s

There is no (88, 165, 1956)-net in base 4, because

1 times m-reduction [i] would yield (88, 164, 1956)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 553 918710 119690 385522 581238 559010 246037 375246 237408 230504 618918 007763 308278 169785 363539 429415 850400 > 4¹⁶⁴ [i]