MinT - Best Known (30−26, 30, s)-Nets in Base 8

Best Known (30−26, 30, s)-Nets in Base 8

(30−26, 30, 25)-Net over F₈ — Constructive and digital

Digital (4, 30, 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]

(30−26, 30, 75)-Net over F₈ — Upper bound on s (digital)

There is no digital (4, 30, 76)-net over F₈, because

2 times m-reduction [i] would yield digital (4, 28, 76)-net over F₈, but
- extracting embedded orthogonal array [i] would yield linear OA(8²⁸, 76, F₈, 24) (dual of [76, 48, 25]-code), but
  - constructionÂ Y1 [i] would yield
    1. linear OA(8²⁷, 34, F₈, 24) (dual of [34, 7, 25]-code), but
      - constructionÂ Y1 [i] would yield
        OA(8²⁶, 28, S₈, 24), but
        the (dual) Plotkin bound shows that MÂ â‰¥ 9 671406 556917 033397 649408 / 25 > 8²⁶ [i]
        OA(8⁷, 34, S₈, 6), but
        discarding factors would yield OA(8⁷, 32, S₈, 6), but
        the linear programming bound shows that MÂ â‰¥ 3784 900608 / 1729 > 8⁷ [i]
    2. linear OA(8⁴⁸, 76, F₈, 42) (dual of [76, 28, 43]-code), but
      - discarding factors / shortening the dual code would yield linear OA(8⁴⁸, 58, F₈, 42) (dual of [58, 10, 43]-code), but
        constructionÂ Y1 [i] would yield
        linear OA(8⁴⁷, 50, F₈, 42) (dual of [50, 3, 43]-code), but
        â€œHi4â€ bound on codes from Brouwerâ€™s database [i]
        OA(8¹⁰, 58, S₈, 8), but
        discarding factors would yield OA(8¹⁰, 57, S₈, 8), but
        the linear programming bound shows that MÂ â‰¥ 1426 626655 027200 / 1 315507 > 8¹⁰ [i]

(30−26, 30, 79)-Net in Base 8 — Upper bound on s

There is no (4, 30, 80)-net in base 8, because

extracting embedded orthogonal array [i] would yield OA(8³⁰, 80, S₈, 26), but
- the linear programming bound shows that MÂ â‰¥ 13 682212 274302 658089 590997 718068 536897 550187 233280 / 10656 691984 486719 330503 > 8³⁰ [i]