MinT - Best Known (251−109, 251, s)-Nets in Base 4

Best Known (251−109, 251, s)-Nets in Base 4

(251−109, 251, 137)-Net over F₄ — Constructive and digital

Digital (142, 251, 137)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (15, 69, 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, 182, 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]

(251−109, 251, 274)-Net over F₄ — Digital

Digital (142, 251, 274)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(251−109, 251, 4238)-Net in Base 4 — Upper bound on s

There is no (142, 251, 4239)-net in base 4, because

1 times m-reduction [i] would yield (142, 250, 4239)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3 291551 136800 323497 557149 049247 117050 664818 996391 726462 453648 128460 313357 239466 462962 187987 593165 959843 814959 501949 749511 130549 358456 236919 205377 870344 > 4²⁵⁰ [i]