MinT - Best Known (4, 4+29, s)-Nets in Base 8

Best Known (4, 4+29, s)-Nets in Base 8

(4, 4+29, 25)-Net over F₈ — Constructive and digital

Digital (4, 33, 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+29, 45)-Net over F₈ — Upper bound on s (digital)

There is no digital (4, 33, 46)-net over F₈, because

1 times m-reduction [i] would yield digital (4, 32, 46)-net over F₈, but
- 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+29, 69)-Net in Base 8 — Upper bound on s

There is no (4, 33, 70)-net in base 8, because

extracting embedded orthogonal array [i] would yield OA(8³³, 70, S₈, 29), but
- the linear programming bound shows that MÂ â‰¥ 27 702182 052296 607762 357735 488830 788159 349604 220894 445568 / 40 071532 500169 293729 282319 > 8³³ [i]