MinT - Best Known (91, 101, s)-Nets in Base 3

Best Known (91, 101, s)-Nets in Base 3

(91, 101, 1677758)-Net over F₃ — Constructive and digital

Digital (91, 101, 1677758)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (5, 10, 38)-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]

(91, 101, 4194339)-Net over F₃ — Digital

Digital (91, 101, 4194339)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹⁰¹, 4194339, F₃, 2, 10) (dual of [(4194339, 2), 8388577, 11]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(3¹⁰, 38, F₃, 2, 5) (dual of [(38, 2), 66, 6]-NRT-code), using
    - extracting embedded OOA [i] based on digital (5, 10, 38)-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]

(91, 101, large)-Net in Base 3 — Upper bound on s

There is no (91, 101, large)-net in base 3, because

8 times m-reduction [i] would yield (91, 93, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]