MinT - Best Known (6, 67, s)-Nets in Base 5

Best Known (6, 67, s)-Nets in Base 5

(6, 67, 21)-Net over F₅ — Constructive and digital

Digital (6, 67, 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]

(6, 67, 38)-Net over F₅ — Upper bound on s (digital)

There is no digital (6, 67, 39)-net over F₅, because

36 times m-reduction [i] would yield digital (6, 31, 39)-net over F₅, but
- extracting embedded orthogonal array [i] would yield linear OA(5³¹, 39, F₅, 25) (dual of [39, 8, 26]-code), but
  - constructionÂ Y1 [i] would yield
    1. linear OA(5³⁰, 33, F₅, 25) (dual of [33, 3, 26]-code), but
      - Griesmer bound [i]
    2. OA(5⁸, 39, S₅, 6), but
      - discarding factors would yield OA(5⁸, 34, S₅, 6), but
        the Rao or (dual) Hamming bound shows that MÂ â‰¥ 392089 > 5⁸ [i]

(6, 67, 40)-Net in Base 5 — Upper bound on s

There is no (6, 67, 41)-net in base 5, because

31 times m-reduction [i] would yield (6, 36, 41)-net in base 5, but
- extracting embedded orthogonal array [i] would yield OA(5³⁶, 41, S₅, 30), but
  - the linear programming bound shows that MÂ â‰¥ 187355 908565 223217 010498 046875 / 10323 > 5³⁶ [i]