MinT - Best Known (47, 62, s)-Nets in Base 3

Best Known (47, 62, s)-Nets in Base 3

(47, 62, 328)-Net over F₃ — Constructive and digital

Digital (47, 62, 328)-net over F₃, using

3² times duplication [i] based on digital (45, 60, 328)-net over F₃, using
- trace code for nets [i] based on digital (0, 15, 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]

(47, 62, 480)-Net over F₃ — Digital

Digital (47, 62, 480)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3⁶², 480, F₃, 15) (dual of [480, 418, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(3⁶², 743, F₃, 15) (dual of [743, 681, 16]-code), using
  - constructionÂ X applied to C([0,7]) âŠ‚ C([0,6]) [i] based on
    1. linear OA(3⁶¹, 730, F₃, 15) (dual of [730, 669, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 730Â | 3¹²âˆ’1, defining interval IÂ =Â [0,7], and minimum distance dÂ â‰¥ |{−7,−6,…,7}|+1Â =Â 16 (BCH-bound) [i]
    2. linear OA(3⁴⁹, 730, F₃, 13) (dual of [730, 681, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 730Â | 3¹²âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (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]

(47, 62, 24296)-Net in Base 3 — Upper bound on s

There is no (47, 62, 24297)-net in base 3, because

1 times m-reduction [i] would yield (47, 61, 24297)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 127192 380175 830513 074511 895851 > 3⁶¹ [i]