MinT - Best Known (50−44, 50, s)-Nets in Base 8

Best Known (50−44, 50, s)-Nets in Base 8

(50−44, 50, 28)-Net over F₈ — Constructive and digital

Digital (6, 50, 28)-net over F₈, using

t-expansion [i] based on digital (5, 50, 28)-net over F₈, using
- net from sequence [i] based on digital (5, 27)-sequence over F₈, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 5 and N(F) ≥ 28, using
    - a function field by SÃ©mirat [i]

(50−44, 50, 33)-Net over F₈ — Digital

Digital (6, 50, 33)-net over F₈, using

net from sequence [i] based on digital (6, 32)-sequence over F₈, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 6 and N(F) ≥ 33, using
  - Drinfeld modules of rank 1 [i]

(50−44, 50, 57)-Net over F₈ — Upper bound on s (digital)

There is no digital (6, 50, 58)-net over F₈, because

2 times m-reduction [i] would yield digital (6, 48, 58)-net over F₈, but
- extracting embedded orthogonal array [i] would yield linear OA(8⁴⁸, 58, F₈, 42) (dual of [58, 10, 43]-code), but
  - constructionÂ Y1 [i] would yield
    1. linear OA(8⁴⁷, 50, F₈, 42) (dual of [50, 3, 43]-code), but
      - â€œHi4â€ bound on codes from Brouwerâ€™s database [i]
    2. 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]

(50−44, 50, 61)-Net in Base 8 — Upper bound on s

There is no (6, 50, 62)-net in base 8, because

extracting embedded orthogonal array [i] would yield OA(8⁵⁰, 62, S₈, 44), but
- the linear programming bound shows that MÂ â‰¥ 146730 379883 110660 278895 346150 216666 324396 659517 161472 / 101 061513 > 8⁵⁰ [i]