MinT - Best Known (215−27, 215, s)-Nets in Base 3

Best Known (215−27, 215, s)-Nets in Base 3

(215−27, 215, 13638)-Net over F₃ — Constructive and digital

Digital (188, 215, 13638)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (4, 17, 12)-net over F₃, using
  - net from sequence [i] based on digital (4, 11)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 4 and N(F) ≥ 12, using
      - an explicitly constructive algebraic function field [i]
2. digital (171, 198, 13626)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3¹⁹⁸, 13626, F₃, 27, 27) (dual of [(13626, 27), 367704, 28]-NRT-code), using
    - OOA 13-folding and stacking with additional row [i] based on linear OA(3¹⁹⁸, 177139, F₃, 27) (dual of [177139, 176941, 28]-code), using
      - discarding factors / shortening the dual code based on linear OA(3¹⁹⁸, 177146, F₃, 27) (dual of [177146, 176948, 28]-code), using
        1 times truncation [i] based on linear OA(3¹⁹⁹, 177147, F₃, 28) (dual of [177147, 176948, 29]-code), using
        an extension C_e(27) of the primitive narrow-sense BCH-code C(I) with length 177146Â = 3¹¹âˆ’1, defining interval IÂ =Â [1,27], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 28 [i]

(215−27, 215, 84075)-Net over F₃ — Digital

Digital (188, 215, 84075)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3²¹⁵, 84075, F₃, 2, 27) (dual of [(84075, 2), 167935, 28]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3²¹⁵, 88615, F₃, 2, 27) (dual of [(88615, 2), 177015, 28]-NRT-code), using
  - OOA 2-folding [i] based on linear OA(3²¹⁵, 177230, F₃, 27) (dual of [177230, 177015, 28]-code), using
    - constructionÂ X applied to C([0,13]) âŠ‚ C([0,9]) [i] based on
      1. linear OA(3¹⁹⁹, 177148, F₃, 27) (dual of [177148, 176949, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 177148Â | 3²²âˆ’1, defining interval IÂ =Â [0,13], and minimum distance dÂ â‰¥ |{−13,−12,…,13}|+1Â =Â 28 (BCH-bound) [i]
      2. linear OA(3¹³³, 177148, F₃, 19) (dual of [177148, 177015, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 177148Â | 3²²âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (BCH-bound) [i]
      3. linear OA(3¹⁶, 82, F₃, 7) (dual of [82, 66, 8]-code), using
        generator matrices provided on Edelâ€™s homepage [i]

(215−27, 215, large)-Net in Base 3 — Upper bound on s

There is no (188, 215, large)-net in base 3, because

25 times m-reduction [i] would yield (188, 190, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]