MinT - Best Known (38−9, 38, s)-Nets in Base 3

Best Known (38−9, 38, s)-Nets in Base 3

(38−9, 38, 328)-Net over F₃ — Constructive and digital

Digital (29, 38, 328)-net over F₃, using

3² times duplication [i] based on digital (27, 36, 328)-net over F₃, using
- trace code for nets [i] based on digital (0, 9, 82)-net over F₈₁, using
  - net from sequence [i] based on digital (0, 81)-sequence over F₈₁, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 0 and N(F) ≥ 82, using
      - the rational function field F₈₁(x) [i]
    - Niederreiter sequence [i]

(38−9, 38, 556)-Net over F₃ — Digital

Digital (29, 38, 556)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3³⁸, 556, F₃, 9) (dual of [556, 518, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(3³⁸, 743, F₃, 9) (dual of [743, 705, 10]-code), using
  - constructionÂ X applied to C([0,4]) âŠ‚ C([0,3]) [i] based on
    1. linear OA(3³⁷, 730, F₃, 9) (dual of [730, 693, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 730Â | 3¹²âˆ’1, defining interval IÂ =Â [0,4], and minimum distance dÂ â‰¥ |{−4,−3,…,4}|+1Â =Â 10 (BCH-bound) [i]
    2. linear OA(3²⁵, 730, F₃, 7) (dual of [730, 705, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 730Â | 3¹²âˆ’1, defining interval IÂ =Â [0,3], and minimum distance dÂ â‰¥ |{−3,−2,…,3}|+1Â =Â 8 (BCH-bound) [i]
    3. linear OA(3¹, 13, F₃, 1) (dual of [13, 12, 2]-code), using
      - discarding factors / shortening the dual code based on linear OA(3¹, s, F₃, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(38−9, 38, 28664)-Net in Base 3 — Upper bound on s

There is no (29, 38, 28665)-net in base 3, because

1 times m-reduction [i] would yield (29, 37, 28665)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 450327 647816 875921 > 3³⁷ [i]