MinT - Best Known (69−56, 69, s)-Nets in Base 4

Best Known (69−56, 69, s)-Nets in Base 4

(69−56, 69, 30)-Net over F₄ — Constructive and digital

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]

(69−56, 69, 33)-Net over F₄ — Digital

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]

(69−56, 69, 58)-Net over F₄ — Upper bound on s (digital)

There is no digital (13, 69, 59)-net over F₄, because

16 times m-reduction [i] would yield digital (13, 53, 59)-net over F₄, but
- extracting embedded orthogonal array [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]

(69−56, 69, 61)-Net in Base 4 — Upper bound on s

There is no (13, 69, 62)-net in base 4, because

13 times m-reduction [i] would yield (13, 56, 62)-net in base 4, but
- extracting embedded orthogonal array [i] would yield OA(4⁵⁶, 62, S₄, 43), but
  - the linear programming bound shows that MÂ â‰¥ 26 916866 914644 546426 302092 970676 977664 / 4675 > 4⁵⁶ [i]