MinT - Best Known (15−11, 15, s)-Nets in Base 4

Best Known (15−11, 15, s)-Nets in Base 4

(15−11, 15, 15)-Net over F₄ — Constructive and digital

Digital (4, 15, 15)-net over F₄, using

net from sequence [i] based on digital (4, 14)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 4 and N(F) ≥ 15, using
  - an explicitly constructive algebraic function field [i]

(15−11, 15, 29)-Net over F₄ — Upper bound on s (digital)

There is no digital (4, 15, 30)-net over F₄, because

1 times m-reduction [i] would yield digital (4, 14, 30)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4¹⁴, 30, F₄, 10) (dual of [30, 16, 11]-code), but
  - constructionÂ Y1 [i] would yield
    1. linear OA(4¹³, 18, F₄, 10) (dual of [18, 5, 11]-code), but
      - â€œLizâ€ bound on codes from Brouwerâ€™s database [i]
    2. linear OA(4¹⁶, 30, F₄, 12) (dual of [30, 14, 13]-code), but
      - discarding factors / shortening the dual code would yield linear OA(4¹⁶, 29, F₄, 12) (dual of [29, 13, 13]-code), but
        constructionÂ Y1 [i] would yield
        linear OA(4¹⁵, 19, F₄, 12) (dual of [19, 4, 13]-code), but
        constructionÂ Y1 [i] would yield
        OA(4¹⁴, 16, S₄, 12), but
        the (dual) Plotkin bound shows that MÂ â‰¥ 4294 967296 / 13 > 4¹⁴ [i]
        linear OA(4⁴, 19, F₄, 3) (dual of [19, 15, 4]-code or 19-cap in PG(3,4)), but
        discarding factors / shortening the dual code would yield linear OA(4⁴, 18, F₄, 3) (dual of [18, 14, 4]-code or 18-cap in PG(3,4)), but
        ovoid cap is optimal [i]
        linear OA(4¹³, 29, F₄, 10) (dual of [29, 16, 11]-code), but
        discarding factors / shortening the dual code would yield linear OA(4¹³, 18, F₄, 10) (dual of [18, 5, 11]-code) (see above)

(15−11, 15, 38)-Net in Base 4 — Upper bound on s

There is no (4, 15, 39)-net in base 4, because

1 times m-reduction [i] would yield (4, 14, 39)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 282 191950 > 4¹⁴ [i]