MinT - Best Known (69, 69+159, s)-Nets in Base 4

Best Known (69, 69+159, s)-Nets in Base 4

(69, 69+159, 66)-Net over F₄ — Constructive and digital

Digital (69, 228, 66)-net over F₄, using

t-expansion [i] based on digital (49, 228, 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]

(69, 69+159, 99)-Net over F₄ — Digital

Digital (69, 228, 99)-net over F₄, using

t-expansion [i] based on digital (61, 228, 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]

(69, 69+159, 478)-Net in Base 4 — Upper bound on s

There is no (69, 228, 479)-net in base 4, because

1 times m-reduction [i] would yield (69, 227, 479)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 49596 156695 603575 949215 251570 072107 809194 146187 321401 551599 336756 858798 202493 891832 027125 984989 640473 693673 320985 336164 002350 294819 339164 > 4²²⁷ [i]