MinT - Best Known (251−167, 251, s)-Nets in Base 4

Best Known (251−167, 251, s)-Nets in Base 4

(251−167, 251, 104)-Net over F₄ — Constructive and digital

Digital (84, 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]

(251−167, 251, 129)-Net over F₄ — Digital

Digital (84, 251, 129)-net over F₄, using

t-expansion [i] based on digital (81, 251, 129)-net over F₄, using
- net from sequence [i] based on digital (81, 128)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 81 and N(F) ≥ 129, using
    - Drinfeld modules of rank 1 [i]

(251−167, 251, 621)-Net in Base 4 — Upper bound on s

There is no (84, 251, 622)-net in base 4, because

1 times m-reduction [i] would yield (84, 250, 622)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3 422869 931657 366277 683151 044000 312461 593374 200999 832594 378624 861808 307747 081166 983638 510028 422288 506105 802471 148344 227839 409827 316638 433384 215823 991148 > 4²⁵⁰ [i]