MinT - Best Known (27−21, 27, s)-Nets in Base 5

Best Known (27−21, 27, s)-Nets in Base 5

(27−21, 27, 21)-Net over F₅ — Constructive and digital

Digital (6, 27, 21)-net over F₅, using

net from sequence [i] based on digital (6, 20)-sequence over F₅, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 6 and N(F) ≥ 21, using
  - an explicitly constructive algebraic function field [i]

(27−21, 27, 63)-Net over F₅ — Upper bound on s (digital)

There is no digital (6, 27, 64)-net over F₅, because

1 times m-reduction [i] would yield digital (6, 26, 64)-net over F₅, but
- extracting embedded orthogonal array [i] would yield linear OA(5²⁶, 64, F₅, 20) (dual of [64, 38, 21]-code), but
  - constructionÂ Y1 [i] would yield
    1. linear OA(5²⁵, 34, F₅, 20) (dual of [34, 9, 21]-code), but
      - residual code [i] would yield linear OA(5⁵, 13, F₅, 4) (dual of [13, 8, 5]-code), but
        â€œBouâ€ bound on codes from Brouwerâ€™s database [i]
    2. linear OA(5³⁸, 64, F₅, 30) (dual of [64, 26, 31]-code), but
      - discarding factors / shortening the dual code would yield linear OA(5³⁸, 61, F₅, 30) (dual of [61, 23, 31]-code), but
        constructionÂ Y1 [i] would yield
        linear OA(5³⁷, 43, F₅, 30) (dual of [43, 6, 31]-code), but
        residual code [i] would yield linear OA(5⁷, 12, F₅, 6) (dual of [12, 5, 7]-code), but
        a result by Dodunekov and Landjev [i]
        OA(5²³, 61, S₅, 18), but
        the linear programming bound shows that MÂ â‰¥ 6321 833259 448409 889221 191406 250000 / 502952 782605 714137 > 5²³ [i]

(27−21, 27, 67)-Net in Base 5 — Upper bound on s

There is no (6, 27, 68)-net in base 5, because

1 times m-reduction [i] would yield (6, 26, 68)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 1 577191 765855 239873 > 5²⁶ [i]