MinT - Best Known (28−24, 28, s)-Nets in Base 7

Best Known (28−24, 28, s)-Nets in Base 7

(28−24, 28, 12)-Net over F₇ — Constructive and digital

Digital (4, 28, 12)-net over F₇, using

net from sequence [i] based on digital (4, 11)-sequence over F₇, using
- Niederreiter sequence [i]

(28−24, 28, 24)-Net over F₇ — Digital

Digital (4, 28, 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]

(28−24, 28, 39)-Net over F₇ — Upper bound on s (digital)

There is no digital (4, 28, 40)-net over F₇, because

extracting embedded orthogonal array [i] would yield linear OA(7²⁸, 40, F₇, 24) (dual of [40, 12, 25]-code), but
- constructionÂ Y1 [i] would yield
  1. linear OA(7²⁷, 30, F₇, 24) (dual of [30, 3, 25]-code), but
    - â€œHi4â€ bound on codes from Brouwerâ€™s database [i]
  2. linear OA(7¹², 40, F₇, 10) (dual of [40, 28, 11]-code), but
    - discarding factors / shortening the dual code would yield linear OA(7¹², 36, F₇, 10) (dual of [36, 24, 11]-code), but
      - constructionÂ Y1 [i] would yield
        linear OA(7¹¹, 15, F₇, 10) (dual of [15, 4, 11]-code), but
        â€œBPMâ€ bound on codes from Brouwerâ€™s database [i]
        linear OA(7²⁴, 36, F₇, 21) (dual of [36, 12, 22]-code), but
        discarding factors / shortening the dual code would yield linear OA(7²⁴, 31, F₇, 21) (dual of [31, 7, 22]-code), but
        residual code [i] would yield OA(7³, 9, S₇, 3), but
        bound for OAs with index unity [i]

(28−24, 28, 65)-Net in Base 7 — Upper bound on s

There is no (4, 28, 66)-net in base 7, because

extracting embedded orthogonal array [i] would yield OA(7²⁸, 66, S₇, 24), but
- the linear programming bound shows that MÂ â‰¥ 16809 890505 936726 165142 292468 059872 923375 / 36279 951798 761879 > 7²⁸ [i]