MinT - Best Known (12, 12+5, s)-Nets in Base 3

Best Known (12, 12+5, s)-Nets in Base 3

(12, 12+5, 231)-Net over F₃ — Constructive and digital

Digital (12, 17, 231)-net over F₃, using

net defined by OOA [i] based on linear OOA(3¹⁷, 231, F₃, 5, 5) (dual of [(231, 5), 1138, 6]-NRT-code), using
- appending kth column [i] based on linear OOA(3¹⁷, 231, F₃, 4, 5) (dual of [(231, 4), 907, 6]-NRT-code), using
  - generalized (u,Â u+v)-construction [i] based on
    1. linear OOA(3¹, 77, F₃, 4, 1) (dual of [(77, 4), 307, 2]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(3¹, s, F₃, 4, 1) with arbitrarily large s, using
        appending 3 arbitrary columns [i] based on linear OA(3¹, s, F₃, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]
    2. linear OOA(3⁵, 77, F₃, 4, 2) (dual of [(77, 4), 303, 3]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(3⁵, 121, F₃, 4, 2) (dual of [(121, 4), 479, 3]-NRT-code), using
        appending 2 arbitrary columns [i] based on linear OOA(3⁵, 121, F₃, 2, 2) (dual of [(121, 2), 237, 3]-NRT-code), using
        appending kth column [i] based on linear OA(3⁵, 121, F₃, 2) (dual of [121, 116, 3]-code), using
        Hamming code H(5,3) [i]
    3. linear OOA(3¹¹, 77, F₃, 4, 5) (dual of [(77, 4), 297, 6]-NRT-code), using
      - extracting embedded OOA [i] based on digital (6, 11, 77)-net over F₃, using
        linear code embedding found in a computer search [i]

(12, 12+5, 255)-Net over F₃ — Digital

Digital (12, 17, 255)-net over F₃, using

net defined by OOA [i] based on linear OOA(3¹⁷, 255, F₃, 5, 5) (dual of [(255, 5), 1258, 6]-NRT-code), using
- appending kth column [i] based on linear OOA(3¹⁷, 255, F₃, 4, 5) (dual of [(255, 4), 1003, 6]-NRT-code), using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁷, 255, F₃, 5) (dual of [255, 238, 6]-code), using
    - (u,Â uâˆ’v,Â u+v+w)-construction [i] based on
      1. linear OA(3¹, 85, F₃, 1) (dual of [85, 84, 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]
      2. linear OA(3⁵, 85, F₃, 2) (dual of [85, 80, 3]-code), using
        discarding factors / shortening the dual code based on linear OA(3⁵, 121, F₃, 2) (dual of [121, 116, 3]-code), using
        Hamming code H(5,3) [i]
      3. linear OA(3¹¹, 85, F₃, 5) (dual of [85, 74, 6]-code), using
        codes constructed by inverting constructionÂ Y₁ [i]

(12, 12+5, 4637)-Net in Base 3 — Upper bound on s

There is no (12, 17, 4638)-net in base 3, because

1 times m-reduction [i] would yield (12, 16, 4638)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 43 049917 > 3¹⁶ [i]