MinT - Best Known (105, 105+41, s)-Nets in Base 9

Best Known (105, 105+41, s)-Nets in Base 9

(105, 105+41, 780)-Net over F₉ — Constructive and digital

Digital (105, 146, 780)-net over F₉, using

t-expansion [i] based on digital (104, 146, 780)-net over F₉, using
- 1 times m-reduction [i] based on digital (104, 147, 780)-net over F₉, using
  - (u,Â u+v)-construction [i] based on
    1. digital (8, 29, 40)-net over F₉, using
      - net from sequence [i] based on digital (8, 39)-sequence over F₉, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 8 and N(F) ≥ 40, using
        a function field by SÃ©mirat [i]
    2. digital (75, 118, 740)-net over F₉, using
      - trace code for nets [i] based on digital (16, 59, 370)-net over F₈₁, using
        net from sequence [i] based on digital (16, 369)-sequence over F₈₁, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 16 and N(F) ≥ 370, using
        a function field by GarcÃa and Quoos [i]

(105, 105+41, 6571)-Net over F₉ — Digital

Digital (105, 146, 6571)-net over F₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(9¹⁴⁶, 6571, F₉, 41) (dual of [6571, 6425, 42]-code), using
- constructionÂ X applied to C([0,20]) âŠ‚ C([0,19]) [i] based on
  1. linear OA(9¹⁴⁵, 6562, F₉, 41) (dual of [6562, 6417, 42]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562Â | 9⁸âˆ’1, defining interval IÂ =Â [0,20], and minimum distance dÂ â‰¥ |{−20,−19,…,20}|+1Â =Â 42 (BCH-bound) [i]
  2. linear OA(9¹³⁷, 6562, F₉, 39) (dual of [6562, 6425, 40]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562Â | 9⁸âˆ’1, defining interval IÂ =Â [0,19], and minimum distance dÂ â‰¥ |{−19,−18,…,19}|+1Â =Â 40 (BCH-bound) [i]
  3. linear OA(9¹, 9, F₉, 1) (dual of [9, 8, 2]-code), using
    - discarding factors / shortening the dual code based on linear OA(9¹, s, F₉, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
      - parity-check code [i]

(105, 105+41, large)-Net in Base 9 — Upper bound on s

There is no (105, 146, large)-net in base 9, because

39 times m-reduction [i] would yield (105, 107, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]