MinT - Best Known (56−42, 56, s)-Nets in Base 4

Best Known (56−42, 56, s)-Nets in Base 4

(56−42, 56, 30)-Net over F₄ — Constructive and digital

Digital (14, 56, 30)-net over F₄, using

t-expansion [i] based on digital (13, 56, 30)-net over F₄, using
- net from sequence [i] based on digital (13, 29)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 13 and N(F) ≥ 30, using
    - F₄ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(56−42, 56, 33)-Net over F₄ — Digital

Digital (14, 56, 33)-net over F₄, using

t-expansion [i] based on digital (13, 56, 33)-net over F₄, using
- net from sequence [i] based on digital (13, 32)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 13 and N(F) ≥ 33, using
    - Drinfeld modules of rank 1 [i]

(56−42, 56, 66)-Net over F₄ — Upper bound on s (digital)

There is no digital (14, 56, 67)-net over F₄, because

2 times m-reduction [i] would yield digital (14, 54, 67)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4⁵⁴, 67, F₄, 40) (dual of [67, 13, 41]-code), but
  - constructionÂ Y1 [i] would yield
    1. 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(4¹³, 67, S₄, 8), but
      - the linear programming bound shows that MÂ â‰¥ 921996 181504 / 13559 > 4¹³ [i]

(56−42, 56, 67)-Net in Base 4 — Upper bound on s

There is no (14, 56, 68)-net in base 4, because

extracting embedded orthogonal array [i] would yield OA(4⁵⁶, 68, S₄, 42), but
- the linear programming bound shows that MÂ â‰¥ 17 363928 619241 647913 609079 468028 268294 176768 / 2650 869375 > 4⁵⁶ [i]