MinT - Best Known (159, 159+58, s)-Nets in Base 4

Best Known (159, 159+58, s)-Nets in Base 4

(159, 159+58, 531)-Net over F₄ — Constructive and digital

Digital (159, 217, 531)-net over F₄, using

11 times m-reduction [i] based on digital (159, 228, 531)-net over F₄, using
- trace code for nets [i] based on digital (7, 76, 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]

(159, 159+58, 576)-Net in Base 4 — Constructive

(159, 217, 576)-net in base 4, using

4¹ times duplication [i] based on (158, 216, 576)-net in base 4, using
- trace code for nets [i] based on (14, 72, 192)-net in base 64, using
  - 5 times m-reduction [i] based on (14, 77, 192)-net in base 64, using
    - base change [i] based on digital (3, 66, 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]

(159, 159+58, 1470)-Net over F₄ — Digital

Digital (159, 217, 1470)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(159, 159+58, 124447)-Net in Base 4 — Upper bound on s

There is no (159, 217, 124448)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 44363 114653 212891 104887 778016 399871 720416 295908 459804 211886 893543 098923 707881 671640 508036 359108 525018 972533 164520 514092 499025 176655 > 4²¹⁷ [i]