MinT - Best Known (17−13, 17, s)-Nets in Base 4

Best Known (17−13, 17, s)-Nets in Base 4

(17−13, 17, 15)-Net over F₄ — Constructive and digital

Digital (4, 17, 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]

(17−13, 17, 28)-Net over F₄ — Upper bound on s (digital)

There is no digital (4, 17, 29)-net over F₄, because

1 times m-reduction [i] would yield digital (4, 16, 29)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4¹⁶, 29, F₄, 12) (dual of [29, 13, 13]-code), but
  - constructionÂ Y1 [i] would yield
    1. 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]
    2. 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), but
        â€œLizâ€ bound on codes from Brouwerâ€™s database [i]

(17−13, 17, 33)-Net in Base 4 — Upper bound on s

There is no (4, 17, 34)-net in base 4, because

extracting embedded OOA [i] would yield OOA(4¹⁷, 34, S₄, 2, 13), but
- the linear programming bound for OOAs shows that MÂ â‰¥ 70769 215362 498168 601183 625441 008362 590676 195407 300483 088955 701062 025508 552704 / 4 097876 886625 380787 777152 266729 752513 546256 867968 750675 889338 368541 > 4¹⁷ [i]