MinT - Best Known (6, 6+3, s)-Nets in Base 4

Best Known (6, 6+3, s)-Nets in Base 4

(6, 6+3, 2110)-Net over F₄ — Constructive and digital

Digital (6, 9, 2110)-net over F₄, using

net defined by OOA [i] based on linear OOA(4⁹, 2110, F₄, 3, 3) (dual of [(2110, 3), 6321, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(4⁹, 2110, F₄, 2, 3) (dual of [(2110, 2), 4211, 4]-NRT-code), using
  - OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(4⁹, 2110, F₄, 3) (dual of [2110, 2101, 4]-code or 2110-cap in PG(8,4)), using
    - caps completed using a computer search [i]

(6, 6+3, 8686)-Net over F₄ — Upper bound on s (digital)

There is no digital (6, 9, 8687)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4⁹, 8687, F₄, 3) (dual of [8687, 8678, 4]-code or 8687-cap in PG(8,4)), but
- removing affine subspaces [i] would yield
  1. linear OA(4⁷, 608, F₄, 3) (dual of [608, 601, 4]-code or 608-cap in PG(6,4)), but
    - 1 times Hill recurrence [i] would yield linear OA(4⁶, 154, F₄, 3) (dual of [154, 148, 4]-code or 154-cap in PG(5,4)), but
      - constructionÂ Y1 [i] 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]
        linear OA(4¹⁴⁸, 154, F₄, 112) (dual of [154, 6, 113]-code), but
        discarding factors / shortening the dual code would yield linear OA(4¹⁴⁸, 153, F₄, 112) (dual of [153, 5, 113]-code), but
        residual code [i] would yield linear OA(4³⁶, 40, F₄, 28) (dual of [40, 4, 29]-code), but
        residual code [i] would yield linear OA(4⁸, 11, F₄, 7) (dual of [11, 3, 8]-code), but
        improvement by Maruta on the Griesmer bound [i]
  2. 1752-cap in AG(7,4), but
    - 3 times the recursive bound from Bierbrauer and Edel [i] would yield 41-cap in AG(4,4), but
      - bound on the size of affine caps [i]
  3. 6329-cap in AG(8,4), but
    - 4 times the recursive bound from Bierbrauer and Edel [i] would yield 41-cap in AG(4,4) (see above)

(6, 6+3, 21844)-Net in Base 4 — Upper bound on s

There is no (6, 9, 21845)-net in base 4, because

extracting embedded orthogonal array [i] would yield OA(4⁹, 21845, S₄, 3), but
- Boseâ€“Bush bound for OAs with strength 3 [i]