MinT - Best Known (5, 42, s)-Nets in Base 8

Best Known (5, 42, s)-Nets in Base 8

(5, 42, 28)-Net over F₈ — Constructive and digital

Digital (5, 42, 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]

(5, 42, 29)-Net over F₈ — Digital

Digital (5, 42, 29)-net over F₈, using

net from sequence [i] based on digital (5, 28)-sequence over F₈, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 5 and N(F) ≥ 29, using
  - a curve from the manYPoints database [i]

(5, 42, 52)-Net over F₈ — Upper bound on s (digital)

There is no digital (5, 42, 53)-net over F₈, because

1 times m-reduction [i] would yield digital (5, 41, 53)-net over F₈, but
- extracting embedded orthogonal array [i] would yield linear OA(8⁴¹, 53, F₈, 36) (dual of [53, 12, 37]-code), but
  - constructionÂ Y1 [i] would yield
    1. linear OA(8⁴⁰, 43, F₈, 36) (dual of [43, 3, 37]-code), but
      - â€œHi4â€ bound on codes from Brouwerâ€™s database [i]
    2. linear OA(8¹², 53, F₈, 10) (dual of [53, 41, 11]-code), but
      - discarding factors / shortening the dual code would yield linear OA(8¹², 48, F₈, 10) (dual of [48, 36, 11]-code), but
        constructionÂ Y1 [i] would yield
        linear OA(8¹¹, 16, F₈, 10) (dual of [16, 5, 11]-code), but
        â€œMPaâ€ bound on codes from Brouwerâ€™s database [i]
        linear OA(8³⁶, 48, F₈, 32) (dual of [48, 12, 33]-code), but
        discarding factors / shortening the dual code would yield linear OA(8³⁶, 43, F₈, 32) (dual of [43, 7, 33]-code), but
        residual code [i] would yield OA(8⁴, 10, S₈, 4), but
        bound for OAs with index unity [i]

(5, 42, 69)-Net in Base 8 — Upper bound on s

There is no (5, 42, 70)-net in base 8, because

extracting embedded orthogonal array [i] would yield OA(8⁴², 70, S₈, 37), but
- the linear programming bound shows that MÂ â‰¥ 20572 057732 082881 665655 466014 666989 963258 358198 549337 866240 / 231 989604 921537 889209 > 8⁴² [i]