MinT - Best Known (3, 3+9, s)-Nets in Base 4

Best Known (3, 3+9, s)-Nets in Base 4

(3, 3+9, 14)-Net over F₄ — Constructive and digital

Digital (3, 12, 14)-net over F₄, using

net from sequence [i] based on digital (3, 13)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 3 and N(F) ≥ 14, using
  - an explicitly constructive algebraic function field [i]

(3, 3+9, 28)-Net over F₄ — Upper bound on s (digital)

There is no digital (3, 12, 29)-net over F₄, because

1 times m-reduction [i] would yield digital (3, 11, 29)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4¹¹, 29, F₄, 8) (dual of [29, 18, 9]-code), but
  - constructionÂ Y1 [i] would yield
    1. 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]
    2. linear OA(4¹⁸, 29, F₄, 14) (dual of [29, 11, 15]-code), but
      - discarding factors / shortening the dual code would yield linear OA(4¹⁸, 24, F₄, 14) (dual of [24, 6, 15]-code), but
        â€œBouâ€ bound on codes from Brouwerâ€™s database [i]

(3, 3+9, 29)-Net in Base 4 — Upper bound on s

There is no (3, 12, 30)-net in base 4, because

extracting embedded OOA [i] would yield OOA(4¹², 30, S₄, 2, 9), but
- the linear programming bound for OOAs shows that MÂ â‰¥ 6 838076 773125 148679 580082 176000 / 405781 347641 306745 802261 > 4¹² [i]