MinT - Best Known (14, 14+45, s)-Nets in Base 4

Best Known (14, 14+45, s)-Nets in Base 4

(14, 14+45, 30)-Net over F₄ — Constructive and digital

Digital (14, 59, 30)-net over F₄, using

t-expansion [i] based on digital (13, 59, 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]

(14, 14+45, 33)-Net over F₄ — Digital

Digital (14, 59, 33)-net over F₄, using

t-expansion [i] based on digital (13, 59, 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]

(14, 14+45, 63)-Net over F₄ — Upper bound on s (digital)

There is no digital (14, 59, 64)-net over F₄, because

1 times m-reduction [i] would yield digital (14, 58, 64)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4⁵⁸, 64, F₄, 44) (dual of [64, 6, 45]-code), but
  - residual code [i] would yield linear OA(4¹⁴, 19, F₄, 11) (dual of [19, 5, 12]-code), but
    - 1 times truncation [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]

(14, 14+45, 66)-Net in Base 4 — Upper bound on s

There is no (14, 59, 67)-net in base 4, because

2 times m-reduction [i] would yield (14, 57, 67)-net in base 4, but
- extracting embedded orthogonal array [i] would yield OA(4⁵⁷, 67, S₄, 43), but
  - the linear programming bound shows that MÂ â‰¥ 978 311804 897698 082457 201996 366333 607936 / 33915 > 4⁵⁷ [i]