MinT - Best Known (109−73, 109, s)-Nets in Base 3

Best Known (109−73, 109, s)-Nets in Base 3

(109−73, 109, 38)-Net over F₃ — Constructive and digital

Digital (36, 109, 38)-net over F₃, using

t-expansion [i] based on digital (32, 109, 38)-net over F₃, using
- net from sequence [i] based on digital (32, 37)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 32 and N(F) ≥ 38, using
    - an explicitly constructive algebraic function field [i]

(109−73, 109, 48)-Net over F₃ — Digital

Digital (36, 109, 48)-net over F₃, using

net from sequence [i] based on digital (36, 47)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 36 and N(F) ≥ 48, using
  - a curve from the manYPoints database [i]

(109−73, 109, 116)-Net over F₃ — Upper bound on s (digital)

There is no digital (36, 109, 117)-net over F₃, because

1 times m-reduction [i] would yield digital (36, 108, 117)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3¹⁰⁸, 117, F₃, 72) (dual of [117, 9, 73]-code), but
  - constructionÂ Y1 [i] would yield
    1. linear OA(3¹⁰⁷, 113, F₃, 72) (dual of [113, 6, 73]-code), but
      - residual code [i] would yield linear OA(3³⁵, 40, F₃, 24) (dual of [40, 5, 25]-code), but
        â€œvE1â€ bound on codes from Brouwerâ€™s database [i]
    2. OA(3⁹, 117, S₃, 4), but
      - discarding factors would yield OA(3⁹, 100, S₃, 4), but
        the Rao or (dual) Hamming bound shows that MÂ â‰¥ 20001 > 3⁹ [i]

(109−73, 109, 117)-Net in Base 3 — Upper bound on s

There is no (36, 109, 118)-net in base 3, because

4 times m-reduction [i] would yield (36, 105, 118)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3¹⁰⁵, 118, S₃, 69), but
  - the linear programming bound shows that MÂ â‰¥ 263 758713 430513 222927 586065 511011 987998 995736 826543 303683 / 1 764070 > 3¹⁰⁵ [i]