MinT - Best Known (165−96, 165, s)-Nets in Base 4

Best Known (165−96, 165, s)-Nets in Base 4

(165−96, 165, 66)-Net over F₄ — Constructive and digital

Digital (69, 165, 66)-net over F₄, using

t-expansion [i] based on digital (49, 165, 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]

(165−96, 165, 99)-Net over F₄ — Digital

Digital (69, 165, 99)-net over F₄, using

t-expansion [i] based on digital (61, 165, 99)-net over F₄, using
- net from sequence [i] based on digital (61, 98)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 61 and N(F) ≥ 99, using
    - Drinfeld modules of rank 1 [i]

(165−96, 165, 694)-Net in Base 4 — Upper bound on s

There is no (69, 165, 695)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2254 466549 589188 071649 591004 855311 379915 983530 215691 567570 085712 494264 920704 013082 781924 930723 144145 > 4¹⁶⁵ [i]