MinT - Best Known (179−27, 179, s)-Nets in Base 3

Best Known (179−27, 179, s)-Nets in Base 3

(179−27, 179, 1525)-Net over F₃ — Constructive and digital

Digital (152, 179, 1525)-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 (135, 162, 1513)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3¹⁶², 1513, F₃, 27, 27) (dual of [(1513, 27), 40689, 28]-NRT-code), using
    - OOA 13-folding and stacking with additional row [i] based on linear OA(3¹⁶², 19670, F₃, 27) (dual of [19670, 19508, 28]-code), using
      - discarding factors / shortening the dual code based on linear OA(3¹⁶², 19682, F₃, 27) (dual of [19682, 19520, 28]-code), using
        1 times truncation [i] based on linear OA(3¹⁶³, 19683, F₃, 28) (dual of [19683, 19520, 29]-code), using
        an extension C_e(27) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 3⁹âˆ’1, defining interval IÂ =Â [1,27], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 28 [i]

(179−27, 179, 12673)-Net over F₃ — Digital

Digital (152, 179, 12673)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁷⁹, 12673, F₃, 27) (dual of [12673, 12494, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹⁷⁹, 19754, F₃, 27) (dual of [19754, 19575, 28]-code), using
  - constructionÂ X applied to C([0,13]) âŠ‚ C([0,9]) [i] based on
    1. linear OA(3¹⁶³, 19684, F₃, 27) (dual of [19684, 19521, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684Â | 3¹⁸âˆ’1, defining interval IÂ =Â [0,13], and minimum distance dÂ â‰¥ |{−13,−12,…,13}|+1Â =Â 28 (BCH-bound) [i]
    2. linear OA(3¹⁰⁹, 19684, F₃, 19) (dual of [19684, 19575, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684Â | 3¹⁸âˆ’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, 70, F₃, 7) (dual of [70, 54, 8]-code), using
      - discarding factors / shortening the dual code based on linear OA(3¹⁶, 82, F₃, 7) (dual of [82, 66, 8]-code), using
        generator matrices provided on Edelâ€™s homepage [i]

(179−27, 179, large)-Net in Base 3 — Upper bound on s

There is no (152, 179, large)-net in base 3, because

25 times m-reduction [i] would yield (152, 154, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]