MinT - Best Known (251−175, 251, s)-Nets in Base 4

Best Known (251−175, 251, s)-Nets in Base 4

(251−175, 251, 104)-Net over F₄ — Constructive and digital

Digital (76, 251, 104)-net over F₄, using

t-expansion [i] based on digital (73, 251, 104)-net over F₄, using
- net from sequence [i] based on digital (73, 103)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 73 and N(F) ≥ 104, using
    - F₆ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(251−175, 251, 112)-Net over F₄ — Digital

Digital (76, 251, 112)-net over F₄, using

t-expansion [i] based on digital (73, 251, 112)-net over F₄, using
- net from sequence [i] based on digital (73, 111)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 73 and N(F) ≥ 112, using
    - Drinfeld modules of rank 1 [i]

(251−175, 251, 525)-Net in Base 4 — Upper bound on s

There is no (76, 251, 526)-net in base 4, because

1 times m-reduction [i] would yield (76, 250, 526)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3 636564 538966 082273 086233 083403 537078 570834 607442 386334 690940 342229 108290 862265 955426 259778 266756 466985 775322 665044 331723 855667 687604 900819 542947 919488 > 4²⁵⁰ [i]