MinT - Best Known (229−62, 229, s)-Nets in Base 4

Best Known (229−62, 229, s)-Nets in Base 4

(229−62, 229, 531)-Net over F₄ — Constructive and digital

Digital (167, 229, 531)-net over F₄, using

11 times m-reduction [i] based on digital (167, 240, 531)-net over F₄, using
- trace code for nets [i] based on digital (7, 80, 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]

(229−62, 229, 576)-Net in Base 4 — Constructive

(167, 229, 576)-net in base 4, using

4¹ times duplication [i] based on (166, 228, 576)-net in base 4, using
- trace code for nets [i] based on (14, 76, 192)-net in base 64, using
  - 1 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]

(229−62, 229, 1455)-Net over F₄ — Digital

Digital (167, 229, 1455)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(229−62, 229, 115955)-Net in Base 4 — Upper bound on s

There is no (167, 229, 115956)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 744334 059221 328591 254506 395390 643662 551706 701366 974067 286722 531785 240078 628978 064530 421158 592919 349802 417656 275969 915921 932318 634719 753784 > 4²²⁹ [i]