MinT - Best Known (23, 23+9, s)-Nets in Base 3

Best Known (23, 23+9, s)-Nets in Base 3

(23, 23+9, 114)-Net over F₃ — Constructive and digital

Digital (23, 32, 114)-net over F₃, using

1 times m-reduction [i] based on digital (23, 33, 114)-net over F₃, using
- trace code for nets [i] based on digital (1, 11, 38)-net over F₂₇, using
  - net from sequence [i] based on digital (1, 37)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 1 and N(F) ≥ 38, using
      - a function field by SÃ©mirat [i]

(23, 23+9, 214)-Net over F₃ — Digital

Digital (23, 32, 214)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3³², 214, F₃, 9) (dual of [214, 182, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(3³², 255, F₃, 9) (dual of [255, 223, 10]-code), using
  - constructionÂ X applied to C([0,4]) âŠ‚ C([0,3]) [i] based on
    1. linear OA(3³¹, 244, F₃, 9) (dual of [244, 213, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 244Â | 3¹⁰âˆ’1, defining interval IÂ =Â [0,4], and minimum distance dÂ â‰¥ |{−4,−3,…,4}|+1Â =Â 10 (BCH-bound) [i]
    2. linear OA(3²¹, 244, F₃, 7) (dual of [244, 223, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 244Â | 3¹⁰âˆ’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, 11, F₃, 1) (dual of [11, 10, 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]

(23, 23+9, 5513)-Net in Base 3 — Upper bound on s

There is no (23, 32, 5514)-net in base 3, because

1 times m-reduction [i] would yield (23, 31, 5514)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 617 842552 286633 > 3³¹ [i]