MinT - Best Known (170−45, 170, s)-Nets in Base 4

Best Known (170−45, 170, s)-Nets in Base 4

(170−45, 170, 531)-Net over F₄ — Constructive and digital

Digital (125, 170, 531)-net over F₄, using

7 times m-reduction [i] based on digital (125, 177, 531)-net over F₄, using
- trace code for nets [i] based on digital (7, 59, 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]

(170−45, 170, 576)-Net in Base 4 — Constructive

(125, 170, 576)-net in base 4, using

4² times duplication [i] based on (123, 168, 576)-net in base 4, using
- trace code for nets [i] based on (11, 56, 192)-net in base 64, using
  - base change [i] based on digital (3, 48, 192)-net over F₁₂₈, using
    - net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 3 and N(F) ≥ 192, using
        a function field by SÃ©mirat [i]

(170−45, 170, 1241)-Net over F₄ — Digital

Digital (125, 170, 1241)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(170−45, 170, 127229)-Net in Base 4 — Upper bound on s

There is no (125, 170, 127230)-net in base 4, because

1 times m-reduction [i] would yield (125, 169, 127230)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 560027 891545 174492 445006 019264 857414 000411 221021 554348 330944 512538 373123 342975 434274 021297 714370 857436 > 4¹⁶⁹ [i]