MinT - Best Known (44−33, 44, s)-Nets in Base 4

Best Known (44−33, 44, s)-Nets in Base 4

(44−33, 44, 27)-Net over F₄ — Constructive and digital

Digital (11, 44, 27)-net over F₄, using

t-expansion [i] based on digital (10, 44, 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]

(44−33, 44, 54)-Net over F₄ — Upper bound on s (digital)

There is no digital (11, 44, 55)-net over F₄, because

1 times m-reduction [i] would yield digital (11, 43, 55)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4⁴³, 55, F₄, 32) (dual of [55, 12, 33]-code), but
  - constructionÂ Y1 [i] would yield
    1. linear OA(4⁴², 47, F₄, 32) (dual of [47, 5, 33]-code), but
      - constructionÂ Y1 [i] would yield
        linear OA(4⁴¹, 44, F₄, 32) (dual of [44, 3, 33]-code), but
        Griesmer bound [i]
        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]
    2. OA(4¹², 55, S₄, 8), but
      - discarding factors would yield OA(4¹², 49, S₄, 8), but
        the Rao or (dual) Hamming bound shows that MÂ â‰¥ 17 670136 > 4¹² [i]

(44−33, 44, 57)-Net in Base 4 — Upper bound on s

There is no (11, 44, 58)-net in base 4, because

extracting embedded orthogonal array [i] would yield OA(4⁴⁴, 58, S₄, 33), but
- the linear programming bound shows that MÂ â‰¥ 1 400381 105133 482617 813310 620882 173952 / 4261 482225 > 4⁴⁴ [i]