MinT - Best Known (135, 135+101, s)-Nets in Base 4

Best Known (135, 135+101, s)-Nets in Base 4

(135, 135+101, 132)-Net over F₄ — Constructive and digital

Digital (135, 236, 132)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (12, 62, 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, 174, 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]

(135, 135+101, 272)-Net over F₄ — Digital

Digital (135, 236, 272)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(135, 135+101, 4346)-Net in Base 4 — Upper bound on s

There is no (135, 236, 4347)-net in base 4, because

1 times m-reduction [i] would yield (135, 235, 4347)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3064 896749 373726 326608 683841 711472 233035 611157 200591 153892 865114 370132 928525 915680 565571 706570 314040 848307 089744 285246 366707 605063 058619 017865 > 4²³⁵ [i]