MinT - Best Known (170, 170+61, s)-Nets in Base 4

Best Known (170, 170+61, s)-Nets in Base 4

(170, 170+61, 531)-Net over F₄ — Constructive and digital

Digital (170, 231, 531)-net over F₄, using

t-expansion [i] based on digital (169, 231, 531)-net over F₄, using
- 12 times m-reduction [i] based on digital (169, 243, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 81, 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]

(170, 170+61, 648)-Net in Base 4 — Constructive

(170, 231, 648)-net in base 4, using

trace code for nets [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]

(170, 170+61, 1637)-Net over F₄ — Digital

Digital (170, 231, 1637)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(170, 170+61, 165728)-Net in Base 4 — Upper bound on s

There is no (170, 231, 165729)-net in base 4, because

1 times m-reduction [i] would yield (170, 230, 165729)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2 977547 410265 081852 878150 236324 919356 836257 305711 577229 844929 082045 585503 967096 043668 591475 149428 618043 337468 225449 122173 361234 194004 838544 > 4²³⁰ [i]