MinT - Best Known (159, 159+88, s)-Nets in Base 4

Best Known (159, 159+88, s)-Nets in Base 4

(159, 159+88, 163)-Net over F₄ — Constructive and digital

Digital (159, 247, 163)-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 (100, 188, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 94, 65)-net over F₁₆, using
    - net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
        the Hermitian function field over F₁₆ [i]

(159, 159+88, 208)-Net in Base 4 — Constructive

(159, 247, 208)-net in base 4, using

3 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+88, 524)-Net over F₄ — Digital

Digital (159, 247, 524)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(159, 159+88, 13753)-Net in Base 4 — Upper bound on s

There is no (159, 247, 13754)-net in base 4, because

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]