MinT - Best Known (221−89, 221, s)-Nets in Base 4

Best Known (221−89, 221, s)-Nets in Base 4

(221−89, 221, 137)-Net over F₄ — Constructive and digital

Digital (132, 221, 137)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (15, 59, 33)-net over F₄, using
  - net from sequence [i] based on digital (15, 32)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 15 and N(F) ≥ 33, using
      - an explicitly constructive algebraic function field [i]
2. digital (73, 162, 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]

(221−89, 221, 312)-Net over F₄ — Digital

Digital (132, 221, 312)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(221−89, 221, 5853)-Net in Base 4 — Upper bound on s

There is no (132, 221, 5854)-net in base 4, because

1 times m-reduction [i] would yield (132, 220, 5854)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2 844948 869874 646331 283878 987920 010930 383926 435641 089761 570860 862042 200362 300115 298469 362653 785547 566464 658668 461166 404177 983962 112306 > 4²²⁰ [i]