MinT - Best Known (21−18, 21, s)-Nets in Base 7

Best Known (21−18, 21, s)-Nets in Base 7

(21−18, 21, 11)-Net over F₇ — Constructive and digital

Digital (3, 21, 11)-net over F₇, using

net from sequence [i] based on digital (3, 10)-sequence over F₇, using
- Niederreiter sequence [i]

(21−18, 21, 20)-Net over F₇ — Digital

Digital (3, 21, 20)-net over F₇, using

net from sequence [i] based on digital (3, 19)-sequence over F₇, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₇ with g(F) = 3 and N(F) ≥ 20, using
  - fields by Serre with genus 3 or 4 [i]

(21−18, 21, 34)-Net over F₇ — Upper bound on s (digital)

There is no digital (3, 21, 35)-net over F₇, because

extracting embedded orthogonal array [i] would yield linear OA(7²¹, 35, F₇, 18) (dual of [35, 14, 19]-code), but
- constructionÂ Y1 [i] would yield
  1. linear OA(7²⁰, 23, F₇, 18) (dual of [23, 3, 19]-code), but
    - â€œHi4â€ bound on codes from Brouwerâ€™s database [i]
  2. linear OA(7¹⁴, 35, F₇, 12) (dual of [35, 21, 13]-code), but
    - discarding factors / shortening the dual code would yield linear OA(7¹⁴, 30, F₇, 12) (dual of [30, 16, 13]-code), but
      - constructionÂ Y1 [i] would yield
        linear OA(7¹³, 16, F₇, 12) (dual of [16, 3, 13]-code), but
        bound for almost-MDS-codes with dimension nÂ =Â 3 [i]
        linear OA(7¹⁶, 30, F₇, 14) (dual of [30, 14, 15]-code), but
        discarding factors / shortening the dual code would yield linear OA(7¹⁶, 24, F₇, 14) (dual of [24, 8, 15]-code), but
        residual code [i] would yield OA(7², 9, S₇, 2), but
        bound for OAs with strength kÂ =Â 2 [i]
        the Rao or (dual) Hamming bound shows that MÂ â‰¥ 55 > 7² [i]

(21−18, 21, 59)-Net in Base 7 — Upper bound on s

There is no (3, 21, 60)-net in base 7, because

the generalized Rao bound for nets shows that 7^mÂ â‰¥ 603404 765742 708073 > 7²¹ [i]