MinT - Best Known (3, 3+23, s)-Nets in Base 8

Best Known (3, 3+23, s)-Nets in Base 8

(3, 3+23, 24)-Net over F₈ — Constructive and digital

Digital (3, 26, 24)-net over F₈, using

net from sequence [i] based on digital (3, 23)-sequence over F₈, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 3 and N(F) ≥ 24, using
  - the Klein quartic over F₈ [i]

(3, 3+23, 51)-Net over F₈ — Upper bound on s (digital)

There is no digital (3, 26, 52)-net over F₈, because

1 times m-reduction [i] would yield digital (3, 25, 52)-net over F₈, but
- extracting embedded orthogonal array [i] would yield linear OA(8²⁵, 52, F₈, 22) (dual of [52, 27, 23]-code), but
  - constructionÂ Y1 [i] would yield
    1. OA(8²⁴, 28, S₈, 22), but
      - the linear programming bound shows that MÂ â‰¥ 41 859056 504156 535174 201344 / 8073 > 8²⁴ [i]
    2. linear OA(8²⁷, 52, F₈, 24) (dual of [52, 25, 25]-code), but
      - discarding factors / shortening the dual code would yield linear OA(8²⁷, 36, F₈, 24) (dual of [36, 9, 25]-code), but
        residual code [i] would yield OA(8³, 11, S₈, 3), but
        1 times truncation [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]

(3, 3+23, 62)-Net in Base 8 — Upper bound on s

There is no (3, 26, 63)-net in base 8, because

extracting embedded orthogonal array [i] would yield OA(8²⁶, 63, S₈, 23), but
- the linear programming bound shows that MÂ â‰¥ 128 285586 204438 392324 581916 736944 603136 / 393 441456 889879 > 8²⁶ [i]