MinT - Best Known (169, 169+59, s)-Nets in Base 4

Best Known (169, 169+59, s)-Nets in Base 4

(169, 169+59, 540)-Net over F₄ — Constructive and digital

Digital (169, 228, 540)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (1, 30, 9)-net over F₄, using
  - net from sequence [i] based on digital (1, 8)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 1 and N(F) ≥ 9, using
      - the Hermitian function field over F₄ [i]
2. digital (139, 198, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 66, 177)-net over F₆₄, using
    - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
        a function field by GarcÃa and Quoos [i]

(169, 169+59, 648)-Net in Base 4 — Constructive

(169, 228, 648)-net in base 4, using

t-expansion [i] based on (168, 228, 648)-net in base 4, using
- trace code for nets [i] based on (16, 76, 216)-net in base 64, using
  - 1 times m-reduction [i] based on (16, 77, 216)-net in base 64, using
    - base change [i] based on digital (5, 66, 216)-net over F₁₂₈, using
      - net from sequence [i] based on digital (5, 215)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 5 and N(F) ≥ 216, using
        a function field by SÃ©mirat [i]

(169, 169+59, 1770)-Net over F₄ — Digital

Digital (169, 228, 1770)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(169, 169+59, 200736)-Net in Base 4 — Upper bound on s

There is no (169, 228, 200737)-net in base 4, because

1 times m-reduction [i] would yield (169, 227, 200737)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 46521 932351 404691 378050 995896 351583 592688 187807 622257 725242 708858 426936 581690 762658 202576 832636 510754 361094 211246 887679 750624 370402 579584 > 4²²⁷ [i]