MinT - Best Known (68−56, 68, s)-Nets in Base 4

Best Known (68−56, 68, s)-Nets in Base 4

(68−56, 68, 28)-Net over F₄ — Constructive and digital

Digital (12, 68, 28)-net over F₄, using

net from sequence [i] based on digital (12, 27)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 12 and N(F) ≥ 28, using
  - an explicitly constructive algebraic function field [i]

(68−56, 68, 29)-Net over F₄ — Digital

Digital (12, 68, 29)-net over F₄, using

net from sequence [i] based on digital (12, 28)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 12 and N(F) ≥ 29, using
  - a curve from the manYPoints database [i]

(68−56, 68, 55)-Net over F₄ — Upper bound on s (digital)

There is no digital (12, 68, 56)-net over F₄, because

16 times m-reduction [i] would yield digital (12, 52, 56)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4⁵², 56, F₄, 40) (dual of [56, 4, 41]-code), but
  - a result by Hill and Landjev [i]
  - improvement by Maruta on the Griesmer bound [i]

(68−56, 68, 57)-Net in Base 4 — Upper bound on s

There is no (12, 68, 58)-net in base 4, because

16 times m-reduction [i] would yield (12, 52, 58)-net in base 4, but
- extracting embedded orthogonal array [i] would yield OA(4⁵², 58, S₄, 40), but
  - the linear programming bound shows that MÂ â‰¥ 44783 560404 862888 296075 530839 523328 / 1927 > 4⁵² [i]