MinT - Best Known (89, 99, s)-Nets in Base 3

Best Known (89, 99, s)-Nets in Base 3

(89, 99, 1677738)-Net over F₃ — Constructive and digital

Digital (89, 99, 1677738)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (3, 8, 18)-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]

(89, 99, 4194319)-Net over F₃ — Digital

Digital (89, 99, 4194319)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3⁹⁹, 4194319, F₃, 2, 10) (dual of [(4194319, 2), 8388539, 11]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(3⁸, 18, F₃, 2, 5) (dual of [(18, 2), 28, 6]-NRT-code), using
    - extracting embedded OOA [i] based on digital (3, 8, 18)-net over F₃, using
      - linear code embedding found in a computer search [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]

(89, 99, large)-Net in Base 3 — Upper bound on s

There is no (89, 99, large)-net in base 3, because

8 times m-reduction [i] would yield (89, 91, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]