MinT - Best Known (61, 61+77, s)-Nets in Base 4

Best Known (61, 61+77, s)-Nets in Base 4

(61, 61+77, 66)-Net over F₄ — Constructive and digital

Digital (61, 138, 66)-net over F₄, using

t-expansion [i] based on digital (49, 138, 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]

(61, 61+77, 99)-Net over F₄ — Digital

Digital (61, 138, 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]

(61, 61+77, 711)-Net in Base 4 — Upper bound on s

There is no (61, 138, 712)-net in base 4, because

1 times m-reduction [i] would yield (61, 137, 712)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 31802 548990 997708 731772 371942 418245 299905 010392 058712 560276 633906 484850 213478 994702 > 4¹³⁷ [i]