MinT - Best Known (222−167, 222, s)-Nets in Base 4

Best Known (222−167, 222, s)-Nets in Base 4

(222−167, 222, 66)-Net over F₄ — Constructive and digital

Digital (55, 222, 66)-net over F₄, using

t-expansion [i] based on digital (49, 222, 66)-net over F₄, using
- net from sequence [i] based on digital (49, 65)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 49 and N(F) ≥ 66, using
    - T₆ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(222−167, 222, 91)-Net over F₄ — Digital

Digital (55, 222, 91)-net over F₄, using

t-expansion [i] based on digital (50, 222, 91)-net over F₄, using
- net from sequence [i] based on digital (50, 90)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 50 and N(F) ≥ 91, using
    - a curve from the manYPoints database [i]

(222−167, 222, 230)-Net over F₄ — Upper bound on s (digital)

There is no digital (55, 222, 231)-net over F₄, because

3 times m-reduction [i] would yield digital (55, 219, 231)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²¹⁹, 231, F₄, 164) (dual of [231, 12, 165]-code), but
  - constructionÂ Y1 [i] would yield
    1. linear OA(4²¹⁸, 225, F₄, 164) (dual of [225, 7, 165]-code), but
      - residual code [i] would yield linear OA(4⁵⁴, 60, F₄, 41) (dual of [60, 6, 42]-code), but
        1 times truncation [i] would yield linear OA(4⁵³, 59, F₄, 40) (dual of [59, 6, 41]-code), but
        residual code [i] would yield linear OA(4¹³, 18, F₄, 10) (dual of [18, 5, 11]-code), but
        â€œLizâ€ bound on codes from Brouwerâ€™s database [i]
    2. OA(4¹², 231, S₄, 6), but
      - discarding factors would yield OA(4¹², 156, S₄, 6), but
        the Rao or (dual) Hamming bound shows that MÂ â‰¥ 16 866019 > 4¹² [i]

(222−167, 222, 359)-Net in Base 4 — Upper bound on s

There is no (55, 222, 360)-net in base 4, because

1 times m-reduction [i] would yield (55, 221, 360)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 13 824784 448380 576958 234270 688228 073429 136610 918216 160375 193498 751583 737301 336534 979195 637651 764666 361399 406357 564889 856811 481765 273482 > 4²²¹ [i]