MinT - Best Known (137, 137+99, s)-Nets in Base 4

Best Known (137, 137+99, s)-Nets in Base 4

(137, 137+99, 137)-Net over F₄ — Constructive and digital

Digital (137, 236, 137)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (15, 64, 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, 172, 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]

(137, 137+99, 290)-Net over F₄ — Digital

Digital (137, 236, 290)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(137, 137+99, 4875)-Net in Base 4 — Upper bound on s

There is no (137, 236, 4876)-net in base 4, because

1 times m-reduction [i] would yield (137, 235, 4876)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3052 920697 675619 931032 210665 586132 563192 747958 864031 231433 030325 887052 854717 753458 380752 592342 549167 764961 647729 731489 207499 784376 915221 591504 > 4²³⁵ [i]