MinT - Best Known (65, 65+127, s)-Nets in Base 4

Best Known (65, 65+127, s)-Nets in Base 4

(65, 65+127, 66)-Net over F₄ — Constructive and digital

Digital (65, 192, 66)-net over F₄, using

t-expansion [i] based on digital (49, 192, 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]

(65, 65+127, 99)-Net over F₄ — Digital

Digital (65, 192, 99)-net over F₄, using

t-expansion [i] based on digital (61, 192, 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]

(65, 65+127, 491)-Net in Base 4 — Upper bound on s

There is no (65, 192, 492)-net in base 4, because

1 times m-reduction [i] would yield (65, 191, 492)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 10 297608 846047 849737 601501 708284 867544 675690 826566 570578 040776 582681 124373 242531 067953 359952 996291 892458 904061 146640 > 4¹⁹¹ [i]