MinT - Best Known (54, 59, s)-Nets in Base 3

Best Known (54, 59, s)-Nets in Base 3

(54, 59, 7174473)-Net over F₃ — Constructive and digital

Digital (54, 59, 7174473)-net over F₃, using

net defined by OOA [i] based on linear OOA(3⁵⁹, 7174473, F₃, 5, 5) (dual of [(7174473, 5), 35872306, 6]-NRT-code), using
- appending kth column [i] based on linear OOA(3⁵⁹, 7174473, F₃, 4, 5) (dual of [(7174473, 4), 28697833, 6]-NRT-code), using
  - generalized (u,Â u+v)-construction [i] based on
    1. linear OOA(3¹, 2391491, F₃, 4, 1) (dual of [(2391491, 4), 9565963, 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(3¹⁵, 2391491, F₃, 4, 2) (dual of [(2391491, 4), 9565949, 3]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(3¹⁵, 7174453, F₃, 4, 2) (dual of [(7174453, 4), 28697797, 3]-NRT-code), using
        appending 2 arbitrary columns [i] based on linear OOA(3¹⁵, 7174453, F₃, 2, 2) (dual of [(7174453, 2), 14348891, 3]-NRT-code), using
        appending kth column [i] based on linear OA(3¹⁵, 7174453, F₃, 2) (dual of [7174453, 7174438, 3]-code), using
        Hamming code H(15,3) [i]
    3. linear OOA(3⁴³, 2391491, F₃, 4, 5) (dual of [(2391491, 4), 9565921, 6]-NRT-code), using
      - OOA 2-folding and stacking with additional row [i] based on linear OA(3⁴³, 4782983, F₃, 5) (dual of [4782983, 4782940, 6]-code), using
        constructionÂ X applied to C_e(4) âŠ‚ C_e(3) [i] based on
        linear OA(3⁴³, 4782969, F₃, 5) (dual of [4782969, 4782926, 6]-code), using an extension C_e(4) of the primitive narrow-sense BCH-code C(I) with length 4782968Â = 3¹⁴âˆ’1, defining interval IÂ =Â [1,4], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]
        linear OA(3²⁹, 4782969, F₃, 4) (dual of [4782969, 4782940, 5]-code), using an extension C_e(3) of the primitive narrow-sense BCH-code C(I) with length 4782968Â = 3¹⁴âˆ’1, defining interval IÂ =Â [1,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]
        linear OA(3⁰, 14, F₃, 0) (dual of [14, 14, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(3⁰, s, F₃, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(54, 59, large)-Net over F₃ — Digital

Digital (54, 59, large)-net over F₃, using

1 times m-reduction [i] based on digital (54, 60, large)-net over F₃, using
- net defined by OOA [i] based on linear OOA(3⁶⁰, large, F₃, 6, 6), using
  - appending kth column [i] based on linear OOA(3⁶⁰, large, F₃, 5, 6), using
    - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3⁶⁰, large, F₃, 6) (dual of [large, large−60, 7]-code), using
      - the primitive narrow-sense BCH-code C(I) with length 14348906Â = 3¹⁵âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]

(54, 59, large)-Net in Base 3 — Upper bound on s

There is no (54, 59, large)-net in base 3, because

3 times m-reduction [i] would yield (54, 56, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]

Best Known (54, 59, s)-Nets in Base 3

(54, 59, 7174473)-Net over F3 — Constructive and digital

(54, 59, large)-Net over F3 — Digital

(54, 59, large)-Net in Base 3 — Upper bound on s

(54, 59, 7174473)-Net over F₃ — Constructive and digital

(54, 59, large)-Net over F₃ — Digital