MinT - Best Known (109−10, 109, s)-Nets in Base 3

Best Known (109−10, 109, s)-Nets in Base 3

(109−10, 109, 1678005)-Net over F₃ — Constructive and digital

Digital (99, 109, 1678005)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (13, 18, 285)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3¹⁸, 285, F₃, 5, 5) (dual of [(285, 5), 1407, 6]-NRT-code), using
    - appending kth column [i] based on linear OOA(3¹⁸, 285, F₃, 4, 5) (dual of [(285, 4), 1122, 6]-NRT-code), using
      - generalized (u,Â u+v)-construction [i] based on
        linear OOA(3¹, 95, F₃, 4, 1) (dual of [(95, 4), 379, 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]
        linear OOA(3⁵, 95, F₃, 4, 2) (dual of [(95, 4), 375, 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]
        linear OOA(3¹², 95, F₃, 4, 5) (dual of [(95, 4), 368, 6]-NRT-code), using
        extracting embedded OOA [i] based on digital (7, 12, 95)-net over F₃, using
        linear code embedding found in a computer search [i]
2. digital (81, 91, 1677720)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3⁹¹, 1677720, F₃, 10, 10) (dual of [(1677720, 10), 16777109, 11]-NRT-code), using
    - OA 5-folding and stacking [i] based on linear OA(3⁹¹, 8388600, F₃, 10) (dual of [8388600, 8388509, 11]-code), using
      - discarding factors / shortening the dual code based on linear OA(3⁹¹, large, F₃, 10) (dual of [large, large−91, 11]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 14348906Â = 3¹⁵âˆ’1, defining interval IÂ =Â [0,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]

(109−10, 109, 5197523)-Net over F₃ — Digital

Digital (99, 109, 5197523)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁰⁹, 5197523, F₃, 10) (dual of [5197523, 5197414, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹⁰⁹, large, F₃, 10) (dual of [large, large−109, 11]-code), using
  - 18 times code embedding in larger space [i] based on linear OA(3⁹¹, large, F₃, 10) (dual of [large, large−91, 11]-code), using
    - the primitive expurgated narrow-sense BCH-code C(I) with length 14348906Â = 3¹⁵âˆ’1, defining interval IÂ =Â [0,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]

(109−10, 109, large)-Net in Base 3 — Upper bound on s

There is no (99, 109, large)-net in base 3, because

8 times m-reduction [i] would yield (99, 101, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]

Best Known (109−10, 109, s)-Nets in Base 3

(109−10, 109, 1678005)-Net over F3 — Constructive and digital

(109−10, 109, 5197523)-Net over F3 — Digital

(109−10, 109, large)-Net in Base 3 — Upper bound on s

(109−10, 109, 1678005)-Net over F₃ — Constructive and digital

(109−10, 109, 5197523)-Net over F₃ — Digital