MinT - Best Known (209, 209+21, s)-Nets in Base 3

Best Known (209, 209+21, s)-Nets in Base 3

(209, 209+21, 838883)-Net over F₃ — Constructive and digital

Digital (209, 230, 838883)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (10, 20, 23)-net over F₃, using
  - 3 times m-reduction [i] based on digital (10, 23, 23)-net over F₃, using
    - a shift-net [i]
2. digital (189, 210, 838860)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3²¹⁰, 838860, F₃, 21, 21) (dual of [(838860, 21), 17615850, 22]-NRT-code), using
    - OOA 10-folding and stacking with additional row [i] based on linear OA(3²¹⁰, 8388601, F₃, 21) (dual of [8388601, 8388391, 22]-code), using
      - discarding factors / shortening the dual code based on linear OA(3²¹⁰, large, F₃, 21) (dual of [large, large−210, 22]-code), using
        the primitive narrow-sense BCH-code C(I) with length 14348906Â = 3¹⁵âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]

(209, 209+21, 4174786)-Net over F₃ — Digital

Digital (209, 230, 4174786)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3²³⁰, 4174786, F₃, 2, 21) (dual of [(4174786, 2), 8349342, 22]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3²³⁰, 4194324, F₃, 2, 21) (dual of [(4194324, 2), 8388418, 22]-NRT-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OOA(3²⁰, 23, F₃, 2, 10) (dual of [(23, 2), 26, 11]-NRT-code), using
      - extracting embedded OOA [i] based on digital (10, 20, 23)-net over F₃, using
        3 times m-reduction [i] based on digital (10, 23, 23)-net over F₃, using
        a shift-net [i]
    2. linear OOA(3²¹⁰, 4194301, F₃, 2, 21) (dual of [(4194301, 2), 8388392, 22]-NRT-code), using
      - OOA 2-folding [i] based on linear OA(3²¹⁰, 8388602, F₃, 21) (dual of [8388602, 8388392, 22]-code), using
        discarding factors / shortening the dual code based on linear OA(3²¹⁰, large, F₃, 21) (dual of [large, large−210, 22]-code), using
        the primitive narrow-sense BCH-code C(I) with length 14348906Â = 3¹⁵âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]

(209, 209+21, large)-Net in Base 3 — Upper bound on s

There is no (209, 230, large)-net in base 3, because

19 times m-reduction [i] would yield (209, 211, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]