MinT - Best Known (211−20, 211, s)-Nets in Base 3

Best Known (211−20, 211, s)-Nets in Base 3

(211−20, 211, 838875)-Net over F₃ — Constructive and digital

Digital (191, 211, 838875)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (5, 15, 15)-net over F₃, using
  - a shift-net [i]
2. digital (176, 196, 838860)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3¹⁹⁶, 838860, F₃, 20, 20) (dual of [(838860, 20), 16777004, 21]-NRT-code), using
    - OA 10-folding and stacking [i] based on linear OA(3¹⁹⁶, 8388600, F₃, 20) (dual of [8388600, 8388404, 21]-code), using
      - discarding factors / shortening the dual code based on linear OA(3¹⁹⁶, large, F₃, 20) (dual of [large, large−196, 21]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 14348906Â = 3¹⁵âˆ’1, defining interval IÂ =Â [0,19], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]

(211−20, 211, 2796216)-Net over F₃ — Digital

Digital (191, 211, 2796216)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3²¹¹, 2796216, F₃, 3, 20) (dual of [(2796216, 3), 8388437, 21]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(3¹⁵, 15, F₃, 3, 10) (dual of [(15, 3), 30, 11]-NRT-code), using
    - extracting embedded OOA [i] based on digital (5, 15, 15)-net over F₃, using
      - a shift-net [i]
  2. linear OOA(3¹⁹⁶, 2796201, F₃, 3, 20) (dual of [(2796201, 3), 8388407, 21]-NRT-code), using
    - OOA 3-folding [i] based on linear OA(3¹⁹⁶, large, F₃, 20) (dual of [large, large−196, 21]-code), using
      - the primitive expurgated narrow-sense BCH-code C(I) with length 14348906Â = 3¹⁵âˆ’1, defining interval IÂ =Â [0,19], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]

(211−20, 211, large)-Net in Base 3 — Upper bound on s

There is no (191, 211, large)-net in base 3, because

18 times m-reduction [i] would yield (191, 193, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]