MinT - Best Known (221−91, 221, s)-Nets in Base 4

Best Known (221−91, 221, s)-Nets in Base 4

(221−91, 221, 132)-Net over F₄ — Constructive and digital

Digital (130, 221, 132)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (12, 57, 28)-net over F₄, using
  - net from sequence [i] based on digital (12, 27)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 12 and N(F) ≥ 28, using
      - an explicitly constructive algebraic function field [i]
2. digital (73, 164, 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]

(221−91, 221, 291)-Net over F₄ — Digital

Digital (130, 221, 291)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(221−91, 221, 5120)-Net in Base 4 — Upper bound on s

There is no (130, 221, 5121)-net in base 4, because

1 times m-reduction [i] would yield (130, 220, 5121)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2 843944 503934 395140 568893 240052 611844 671996 935704 068349 428177 013378 143195 346737 716285 107673 541679 023809 579255 621252 723869 603974 619136 > 4²²⁰ [i]