MinT - Best Known (225−159, 225, s)-Nets in Base 4

Best Known (225−159, 225, s)-Nets in Base 4

(225−159, 225, 66)-Net over F₄ — Constructive and digital

Digital (66, 225, 66)-net over F₄, using

t-expansion [i] based on digital (49, 225, 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]

(225−159, 225, 99)-Net over F₄ — Digital

Digital (66, 225, 99)-net over F₄, using

t-expansion [i] based on digital (61, 225, 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]

(225−159, 225, 450)-Net in Base 4 — Upper bound on s

There is no (66, 225, 451)-net in base 4, because

1 times m-reduction [i] would yield (66, 224, 451)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 731 356579 936880 519539 437217 249418 287705 638244 985732 756866 099626 075567 021802 935683 821300 691551 790797 574665 557525 467744 902086 873789 079552 > 4²²⁴ [i]