MinT - Best Known (53, 53+177, s)-Nets in Base 4

Best Known (53, 53+177, s)-Nets in Base 4

(53, 53+177, 66)-Net over F₄ — Constructive and digital

Digital (53, 230, 66)-net over F₄, using

t-expansion [i] based on digital (49, 230, 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]

(53, 53+177, 91)-Net over F₄ — Digital

Digital (53, 230, 91)-net over F₄, using

t-expansion [i] based on digital (50, 230, 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]

(53, 53+177, 219)-Net over F₄ — Upper bound on s (digital)

There is no digital (53, 230, 220)-net over F₄, because

17 times m-reduction [i] would yield digital (53, 213, 220)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²¹³, 220, F₄, 160) (dual of [220, 7, 161]-code), but
  - residual code [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]

(53, 53+177, 226)-Net in Base 4 — Upper bound on s

There is no (53, 230, 227)-net in base 4, because

7 times m-reduction [i] would yield (53, 223, 227)-net in base 4, but
- extracting embedded orthogonal array [i] would yield OA(4²²³, 227, S₄, 170), but
  - the (dual) Plotkin bound shows that MÂ â‰¥ 11629 419588 729710 248789 180926 208072 549658 261770 997088 964503 843186 890228 609814 366773 219056 811420 217048 972200 345700 258846 936553 626057 834496 / 57 > 4²²³ [i]