MinT - Best Known (107−10, 107, s)-Nets in Base 3

Best Known (107−10, 107, s)-Nets in Base 3

(107−10, 107, 1677877)-Net over F₃ — Constructive and digital

Digital (97, 107, 1677877)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (11, 16, 157)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3¹⁶, 157, F₃, 5, 5) (dual of [(157, 5), 769, 6]-NRT-code), using
    - appending kth column [i] based on linear OOA(3¹⁶, 157, F₃, 4, 5) (dual of [(157, 4), 612, 6]-NRT-code), using
      - generalized (u,Â u+v)-construction [i] based on
        linear OOA(3¹, 40, F₃, 4, 1) (dual of [(40, 4), 159, 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(3⁴, 40, F₃, 4, 2) (dual of [(40, 4), 156, 3]-NRT-code), using
        appending 2 arbitrary columns [i] based on linear OOA(3⁴, 40, F₃, 2, 2) (dual of [(40, 2), 76, 3]-NRT-code), using
        appending kth column [i] based on linear OA(3⁴, 40, F₃, 2) (dual of [40, 36, 3]-code), using
        Hamming code H(4,3) [i]
        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]
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]

(107−10, 107, 4194520)-Net over F₃ — Digital

Digital (97, 107, 4194520)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹⁰⁷, 4194520, F₃, 2, 10) (dual of [(4194520, 2), 8388933, 11]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(3¹⁶, 219, F₃, 2, 5) (dual of [(219, 2), 422, 6]-NRT-code), using
    - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁶, 219, F₃, 5) (dual of [219, 203, 6]-code), using
      - discarding factors / shortening the dual code based on linear OA(3¹⁶, 242, F₃, 5) (dual of [242, 226, 6]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 242Â = 3⁵âˆ’1, defining interval IÂ =Â [0,4], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
  2. linear OOA(3⁹¹, 4194301, F₃, 2, 10) (dual of [(4194301, 2), 8388511, 11]-NRT-code), using
    - OOA 2-folding [i] based on linear OA(3⁹¹, 8388602, F₃, 10) (dual of [8388602, 8388511, 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]

(107−10, 107, large)-Net in Base 3 — Upper bound on s

There is no (97, 107, large)-net in base 3, because

8 times m-reduction [i] would yield (97, 99, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]

Best Known (107−10, 107, s)-Nets in Base 3

(107−10, 107, 1677877)-Net over F3 — Constructive and digital

(107−10, 107, 4194520)-Net over F3 — Digital

(107−10, 107, large)-Net in Base 3 — Upper bound on s

(107−10, 107, 1677877)-Net over F₃ — Constructive and digital

(107−10, 107, 4194520)-Net over F₃ — Digital