MinT - Best Known (213−151, 213, s)-Nets in Base 4

Best Known (213−151, 213, s)-Nets in Base 4

(213−151, 213, 66)-Net over F₄ — Constructive and digital

Digital (62, 213, 66)-net over F₄, using

t-expansion [i] based on digital (49, 213, 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]

(213−151, 213, 99)-Net over F₄ — Digital

Digital (62, 213, 99)-net over F₄, using

t-expansion [i] based on digital (61, 213, 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]

(213−151, 213, 423)-Net in Base 4 — Upper bound on s

There is no (62, 213, 424)-net in base 4, because

1 times m-reduction [i] would yield (62, 212, 424)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 49 642464 343915 668921 425438 295646 665162 462790 888983 844609 703520 530921 500697 282850 440569 535931 170685 938706 335869 051252 851044 737768 > 4²¹² [i]