MinT - Best Known (159, 159+89, s)-Nets in Base 4

Best Known (159, 159+89, s)-Nets in Base 4

(159, 159+89, 160)-Net over F₄ — Constructive and digital

Digital (159, 248, 160)-net over F₄, using

t-expansion [i] based on digital (157, 248, 160)-net over F₄, using
- 11 times m-reduction [i] based on digital (157, 259, 160)-net over F₄, using
  - (u,Â u+v)-construction [i] based on
    1. digital (33, 84, 56)-net over F₄, using
      - net from sequence [i] based on digital (33, 55)-sequence over F₄, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 33 and N(F) ≥ 56, using
        F₅ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]
    2. digital (73, 175, 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]

(159, 159+89, 208)-Net in Base 4 — Constructive

(159, 248, 208)-net in base 4, using

2 times m-reduction [i] based on (159, 250, 208)-net in base 4, using
- trace code for nets [i] based on (34, 125, 104)-net in base 16, using
  - base change [i] based on digital (9, 100, 104)-net over F₃₂, using
    - net from sequence [i] based on digital (9, 103)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 9 and N(F) ≥ 104, using
        a function field by SÃ©mirat [i]

(159, 159+89, 512)-Net over F₄ — Digital

Digital (159, 248, 512)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(159, 159+89, 13753)-Net in Base 4 — Upper bound on s

There is no (159, 248, 13754)-net in base 4, because

1 times m-reduction [i] would yield (159, 247, 13754)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 51255 944892 598552 450075 003467 640100 912991 252181 213089 780613 320058 172429 503792 959405 528861 921010 051054 024075 985435 396276 801881 345469 762005 298717 118724 > 4²⁴⁷ [i]