MinT - Best Known (56−51, 56, s)-Nets in Base 7

Best Known (56−51, 56, s)-Nets in Base 7

(56−51, 56, 13)-Net over F₇ — Constructive and digital

Digital (5, 56, 13)-net over F₇, using

net from sequence [i] based on digital (5, 12)-sequence over F₇, using
- Niederreiter sequence [i]

(56−51, 56, 24)-Net over F₇ — Digital

Digital (5, 56, 24)-net over F₇, using

t-expansion [i] based on digital (4, 56, 24)-net over F₇, using
- net from sequence [i] based on digital (4, 23)-sequence over F₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₇ with g(F) = 4 and N(F) ≥ 24, using
    - an algebraic function field over â„¤₇ by Yeo [i]

(56−51, 56, 44)-Net over F₇ — Upper bound on s (digital)

There is no digital (5, 56, 45)-net over F₇, because

21 times m-reduction [i] would yield digital (5, 35, 45)-net over F₇, but
- extracting embedded orthogonal array [i] would yield linear OA(7³⁵, 45, F₇, 30) (dual of [45, 10, 31]-code), but
  - constructionÂ Y1 [i] would yield
    1. linear OA(7³⁴, 37, F₇, 30) (dual of [37, 3, 31]-code), but
      - â€œHi4â€ bound on codes from Brouwerâ€™s database [i]
    2. linear OA(7¹⁰, 45, F₇, 8) (dual of [45, 35, 9]-code), but
      - discarding factors / shortening the dual code would yield linear OA(7¹⁰, 42, F₇, 8) (dual of [42, 32, 9]-code), but
        constructionÂ Y1 [i] would yield
        linear OA(7⁹, 14, F₇, 8) (dual of [14, 5, 9]-code), but
        â€œMPaâ€ bound on codes from Brouwerâ€™s database [i]
        linear OA(7³², 42, F₇, 28) (dual of [42, 10, 29]-code), but
        discarding factors / shortening the dual code would yield linear OA(7³², 38, F₇, 28) (dual of [38, 6, 29]-code), but
        residual code [i] would yield linear OA(7⁴, 9, F₇, 4) (dual of [9, 5, 5]-code or 9-arc in PG(3,7)), but
        bound for MDS codes [i]

(56−51, 56, 45)-Net in Base 7 — Upper bound on s

There is no (5, 56, 46)-net in base 7, because

15 times m-reduction [i] would yield (5, 41, 46)-net in base 7, but
- extracting embedded orthogonal array [i] would yield OA(7⁴¹, 46, S₇, 36), but
  - the linear programming bound shows that MÂ â‰¥ 68 946139 584883 864057 594001 122656 706829 / 1517 > 7⁴¹ [i]