MinT - Best Known (245−173, 245, s)-Nets in Base 4

Best Known (245−173, 245, s)-Nets in Base 4

(245−173, 245, 66)-Net over F₄ — Constructive and digital

Digital (72, 245, 66)-net over F₄, using

t-expansion [i] based on digital (49, 245, 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]

(245−173, 245, 105)-Net over F₄ — Digital

Digital (72, 245, 105)-net over F₄, using

t-expansion [i] based on digital (70, 245, 105)-net over F₄, using
- net from sequence [i] based on digital (70, 104)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 70 and N(F) ≥ 105, using
    - Drinfeld modules of rank 1 [i]

(245−173, 245, 490)-Net in Base 4 — Upper bound on s

There is no (72, 245, 491)-net in base 4, because

1 times m-reduction [i] would yield (72, 244, 491)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 832 834003 394533 105767 740793 373900 091786 523676 490697 102675 456635 255619 440590 139564 468164 493466 639883 415918 115623 895600 198359 222653 511649 941188 610380 > 4²⁴⁴ [i]