MinT - Best Known (204−135, 204, s)-Nets in Base 4

Best Known (204−135, 204, s)-Nets in Base 4

(204−135, 204, 66)-Net over F₄ — Constructive and digital

Digital (69, 204, 66)-net over F₄, using

t-expansion [i] based on digital (49, 204, 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]

(204−135, 204, 99)-Net over F₄ — Digital

Digital (69, 204, 99)-net over F₄, using

t-expansion [i] based on digital (61, 204, 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]

(204−135, 204, 519)-Net in Base 4 — Upper bound on s

There is no (69, 204, 520)-net in base 4, because

1 times m-reduction [i] would yield (69, 203, 520)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 166 512860 714218 829109 355891 344686 520315 064402 031087 017275 418913 728578 647861 985788 601126 984900 864895 433066 549695 436939 304472 > 4²⁰³ [i]