MinT - Best Known (68−58, 68, s)-Nets in Base 5

Best Known (68−58, 68, s)-Nets in Base 5

(68−58, 68, 26)-Net over F₅ — Constructive and digital

Digital (10, 68, 26)-net over F₅, using

t-expansion [i] based on digital (9, 68, 26)-net over F₅, using
- net from sequence [i] based on digital (9, 25)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 9 and N(F) ≥ 26, using
    - an explicitly constructive algebraic function field [i]

(68−58, 68, 27)-Net over F₅ — Digital

Digital (10, 68, 27)-net over F₅, using

net from sequence [i] based on digital (10, 26)-sequence over F₅, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 10 and N(F) ≥ 27, using
  - algebraic function fields over â„¤₅ by Niederreiter/Xing [i]

(68−58, 68, 58)-Net over F₅ — Upper bound on s (digital)

There is no digital (10, 68, 59)-net over F₅, because

18 times m-reduction [i] would yield digital (10, 50, 59)-net over F₅, but
- extracting embedded orthogonal array [i] would yield linear OA(5⁵⁰, 59, F₅, 40) (dual of [59, 9, 41]-code), but
  - constructionÂ Y1 [i] would yield
    1. linear OA(5⁴⁹, 53, F₅, 40) (dual of [53, 4, 41]-code), but
      - residual code [i] would yield linear OA(5⁹, 12, F₅, 8) (dual of [12, 3, 9]-code), but
        bound for almost-MDS-codes with dimension nÂ =Â 3 [i]
    2. OA(5⁹, 59, S₅, 6), but
      - discarding factors would yield OA(5⁹, 58, S₅, 6), but
        the Rao or (dual) Hamming bound shows that MÂ â‰¥ 2 001465 > 5⁹ [i]

(68−58, 68, 59)-Net in Base 5 — Upper bound on s

There is no (10, 68, 60)-net in base 5, because

14 times m-reduction [i] would yield (10, 54, 60)-net in base 5, but
- extracting embedded orthogonal array [i] would yield OA(5⁵⁴, 60, S₅, 44), but
  - the linear programming bound shows that MÂ â‰¥ 6938 893903 907228 377647 697925 567626 953125 / 117 > 5⁵⁴ [i]