MinT - Best Known (52−42, 52, s)-Nets in Base 4

Best Known (52−42, 52, s)-Nets in Base 4

(52−42, 52, 27)-Net over F₄ — Constructive and digital

Digital (10, 52, 27)-net over F₄, using

net from sequence [i] based on digital (10, 26)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 10 and N(F) ≥ 27, using
  - an explicitly constructive algebraic function field [i]

(52−42, 52, 46)-Net over F₄ — Upper bound on s (digital)

There is no digital (10, 52, 47)-net over F₄, because

10 times m-reduction [i] would yield digital (10, 42, 47)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4⁴², 47, F₄, 32) (dual of [47, 5, 33]-code), but
  - constructionÂ Y1 [i] would yield
    1. linear OA(4⁴¹, 44, F₄, 32) (dual of [44, 3, 33]-code), but
      - Griesmer bound [i]
    2. linear OA(4⁵, 47, F₄, 3) (dual of [47, 42, 4]-code or 47-cap in PG(4,4)), but
      - discarding factors / shortening the dual code would yield linear OA(4⁵, 42, F₄, 3) (dual of [42, 37, 4]-code or 42-cap in PG(4,4)), but
        41 is the largest size of a cap in PG(4,Â 4) [i]

(52−42, 52, 49)-Net in Base 4 — Upper bound on s

There is no (10, 52, 50)-net in base 4, because

8 times m-reduction [i] would yield (10, 44, 50)-net in base 4, but
- extracting embedded orthogonal array [i] would yield OA(4⁴⁴, 50, S₄, 34), but
  - the linear programming bound shows that MÂ â‰¥ 713053 462628 379038 341895 553024 / 2009 > 4⁴⁴ [i]