MinT - Best Known (4, 4+28, s)-Nets in Base 8

Best Known (4, 4+28, s)-Nets in Base 8

(4, 4+28, 25)-Net over F₈ — Constructive and digital

Digital (4, 32, 25)-net over F₈, using

net from sequence [i] based on digital (4, 24)-sequence over F₈, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 4 and N(F) ≥ 25, using
  - a function field by SÃ©mirat [i]

(4, 4+28, 45)-Net over F₈ — Upper bound on s (digital)

There is no digital (4, 32, 46)-net over F₈, because

extracting embedded orthogonal array [i] would yield linear OA(8³², 46, F₈, 28) (dual of [46, 14, 29]-code), but
- constructionÂ Y1 [i] would yield
  1. linear OA(8³¹, 34, F₈, 28) (dual of [34, 3, 29]-code), but
    - â€œMasâ€ bound on codes from Brouwerâ€™s database [i]
  2. linear OA(8¹⁴, 46, F₈, 12) (dual of [46, 32, 13]-code), but
    - discarding factors / shortening the dual code would yield linear OA(8¹⁴, 32, F₈, 12) (dual of [32, 18, 13]-code), but
      - constructionÂ Y1 [i] would yield
        linear OA(8¹³, 16, F₈, 12) (dual of [16, 3, 13]-code), but
        â€œHi4â€ bound on codes from Brouwerâ€™s database [i]
        linear OA(8¹⁸, 32, F₈, 16) (dual of [32, 14, 17]-code), but
        discarding factors / shortening the dual code would yield linear OA(8¹⁸, 27, F₈, 16) (dual of [27, 9, 17]-code), but
        residual code [i] would yield OA(8², 10, S₈, 2), but
        bound for OAs with strength kÂ =Â 2 [i]
        the Rao or (dual) Hamming bound shows that MÂ â‰¥ 71 > 8² [i]

(4, 4+28, 73)-Net in Base 8 — Upper bound on s

There is no (4, 32, 74)-net in base 8, because

extracting embedded orthogonal array [i] would yield OA(8³², 74, S₈, 28), but
- the linear programming bound shows that MÂ â‰¥ 206 202650 112849 827091 865089 526403 236327 467782 242304 / 2479 768644 710917 804639 > 8³² [i]