MinT - Best Known (223−157, 223, s)-Nets in Base 4

Best Known (223−157, 223, s)-Nets in Base 4

(223−157, 223, 66)-Net over F₄ — Constructive and digital

Digital (66, 223, 66)-net over F₄, using

t-expansion [i] based on digital (49, 223, 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]

(223−157, 223, 99)-Net over F₄ — Digital

Digital (66, 223, 99)-net over F₄, using

t-expansion [i] based on digital (61, 223, 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]

(223−157, 223, 453)-Net in Base 4 — Upper bound on s

There is no (66, 223, 454)-net in base 4, because

1 times m-reduction [i] would yield (66, 222, 454)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 52 779684 025382 501311 139499 373646 988395 451099 809855 180069 335087 307231 261412 820995 331802 718546 806467 807017 874448 047321 608391 193683 609024 > 4²²² [i]