MinT - Best Known (13, 13+40, s)-Nets in Base 4

Best Known (13, 13+40, s)-Nets in Base 4

(13, 13+40, 30)-Net over F₄ — Constructive and digital

Digital (13, 53, 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]

(13, 13+40, 33)-Net over F₄ — Digital

Digital (13, 53, 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]

(13, 13+40, 58)-Net over F₄ — Upper bound on s (digital)

There is no digital (13, 53, 59)-net over F₄, because

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]

(13, 13+40, 63)-Net in Base 4 — Upper bound on s

There is no (13, 53, 64)-net in base 4, because

1 times m-reduction [i] would yield (13, 52, 64)-net in base 4, but
- extracting embedded orthogonal array [i] would yield OA(4⁵², 64, S₄, 39), but
  - the linear programming bound shows that MÂ â‰¥ 2 533840 866964 995882 722882 208659 406848 / 104975 > 4⁵² [i]