MinT - Best Known (135, 135+49, s)-Nets in Base 4

Best Known (135, 135+49, s)-Nets in Base 4

(135, 135+49, 531)-Net over F₄ — Constructive and digital

Digital (135, 184, 531)-net over F₄, using

8 times m-reduction [i] based on digital (135, 192, 531)-net over F₄, using
- trace code for nets [i] based on digital (7, 64, 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]

(135, 135+49, 576)-Net in Base 4 — Constructive

(135, 184, 576)-net in base 4, using

4¹ times duplication [i] based on (134, 183, 576)-net in base 4, using
- trace code for nets [i] based on (12, 61, 192)-net in base 64, using
  - 2 times m-reduction [i] based on (12, 63, 192)-net in base 64, using
    - base change [i] based on digital (3, 54, 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]

(135, 135+49, 1293)-Net over F₄ — Digital

Digital (135, 184, 1293)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(135, 135+49, 127315)-Net in Base 4 — Upper bound on s

There is no (135, 184, 127316)-net in base 4, because

1 times m-reduction [i] would yield (135, 183, 127316)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 150 311531 515017 966549 007520 674332 099592 863243 195826 715875 574028 863699 800893 579239 713628 638268 672171 325640 137064 > 4¹⁸³ [i]