MinT - Best Known (197−135, 197, s)-Nets in Base 4

Best Known (197−135, 197, s)-Nets in Base 4

(197−135, 197, 66)-Net over F₄ — Constructive and digital

Digital (62, 197, 66)-net over F₄, using

t-expansion [i] based on digital (49, 197, 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]

(197−135, 197, 99)-Net over F₄ — Digital

Digital (62, 197, 99)-net over F₄, using

t-expansion [i] based on digital (61, 197, 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]

(197−135, 197, 442)-Net in Base 4 — Upper bound on s

There is no (62, 197, 443)-net in base 4, because

1 times m-reduction [i] would yield (62, 196, 443)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 10201 520221 616632 796934 614196 708025 300220 659450 152219 356701 188838 901636 342967 325532 814712 940465 882281 824422 819597 668888 > 4¹⁹⁶ [i]