MinT - Best Known (69−55, 69, s)-Nets in Base 4

Best Known (69−55, 69, s)-Nets in Base 4

(69−55, 69, 30)-Net over F₄ — Constructive and digital

Digital (14, 69, 30)-net over F₄, using

t-expansion [i] based on digital (13, 69, 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]

(69−55, 69, 33)-Net over F₄ — Digital

Digital (14, 69, 33)-net over F₄, using

t-expansion [i] based on digital (13, 69, 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]

(69−55, 69, 63)-Net over F₄ — Upper bound on s (digital)

There is no digital (14, 69, 64)-net over F₄, because

11 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]

(69−55, 69, 65)-Net in Base 4 — Upper bound on s

There is no (14, 69, 66)-net in base 4, because

9 times m-reduction [i] would yield (14, 60, 66)-net in base 4, but
- extracting embedded orthogonal array [i] would yield OA(4⁶⁰, 66, S₄, 46), but
  - the linear programming bound shows that MÂ â‰¥ 340 282366 920938 463463 374607 431768 211456 / 235 > 4⁶⁰ [i]