MinT - Best Known (9−7, 9, s)-Nets in Base 4

Best Known (9−7, 9, s)-Nets in Base 4

(9−7, 9, 10)-Net over F₄ — Constructive and digital

Digital (2, 9, 10)-net over F₄, using

net from sequence [i] based on digital (2, 9)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 2 and N(F) ≥ 10, using
  - an explicitly constructive algebraic function field [i]

(9−7, 9, 17)-Net over F₄ — Upper bound on s (digital)

There is no digital (2, 9, 18)-net over F₄, because

1 times m-reduction [i] would yield digital (2, 8, 18)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4⁸, 18, F₄, 6) (dual of [18, 10, 7]-code), but
  - constructionÂ Y1 [i] would yield
    1. linear OA(4⁷, 10, F₄, 6) (dual of [10, 3, 7]-code), but
      - bound for almost-MDS-codes with dimension nÂ =Â 3 [i]
    2. linear OA(4¹⁰, 18, F₄, 8) (dual of [18, 8, 9]-code), but
      - discarding factors / shortening the dual code would yield linear OA(4¹⁰, 15, F₄, 8) (dual of [15, 5, 9]-code), but
        residual code [i] would yield OA(4², 6, S₄, 2), but
        bound for OAs with strength kÂ =Â 2 [i]
        the Rao or (dual) Hamming bound shows that MÂ â‰¥ 19 > 4² [i]

(9−7, 9, 21)-Net in Base 4 — Upper bound on s

There is no (2, 9, 22)-net in base 4, because

1 times m-reduction [i] would yield (2, 8, 22)-net in base 4, but
- extracting embedded OOA [i] would yield OOA(4⁸, 22, S₄, 2, 6), but
  - the linear programming bound for OOAs shows that MÂ â‰¥ 61553 303552 / 938313 > 4⁸ [i]