MinT - Best Known (251−133, 251, s)-Nets in Base 4

Best Known (251−133, 251, s)-Nets in Base 4

(251−133, 251, 130)-Net over F₄ — Constructive and digital

Digital (118, 251, 130)-net over F₄, using

t-expansion [i] based on digital (105, 251, 130)-net over F₄, using
- net from sequence [i] based on digital (105, 129)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 105 and N(F) ≥ 130, using
    - T₇ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(251−133, 251, 168)-Net over F₄ — Digital

Digital (118, 251, 168)-net over F₄, using

t-expansion [i] based on digital (115, 251, 168)-net over F₄, using
- net from sequence [i] based on digital (115, 167)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 115 and N(F) ≥ 168, using
    - Drinfeld modules of rank 1 [i]

(251−133, 251, 1562)-Net in Base 4 — Upper bound on s

There is no (118, 251, 1563)-net in base 4, because

1 times m-reduction [i] would yield (118, 250, 1563)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3 354547 486905 248102 917922 028417 526637 857470 567447 987729 830696 756374 046984 761432 527646 306347 139863 326932 706676 532319 234567 356566 842177 887801 349649 222760 > 4²⁵⁰ [i]