MinT - Best Known (252−147, 252, s)-Nets in Base 4

Best Known (252−147, 252, s)-Nets in Base 4

(252−147, 252, 130)-Net over F₄ — Constructive and digital

Digital (105, 252, 130)-net over F₄, using

net from sequence [i] based on digital (105, 129)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 105 and N(F) ≥ 130, using
  - T₇ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(252−147, 252, 144)-Net over F₄ — Digital

Digital (105, 252, 144)-net over F₄, using

t-expansion [i] based on digital (91, 252, 144)-net over F₄, using
- net from sequence [i] based on digital (91, 143)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 91 and N(F) ≥ 144, using
    - Drinfeld modules of rank 1 [i]

(252−147, 252, 1037)-Net in Base 4 — Upper bound on s

There is no (105, 252, 1038)-net in base 4, because

1 times m-reduction [i] would yield (105, 251, 1038)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 13 109464 638603 241010 893604 611628 315427 256393 418298 755042 301402 277908 123149 067530 500012 253656 823395 816588 898919 701540 159546 814146 673812 729105 461750 183600 > 4²⁵¹ [i]