MinT - Best Known (144−19, 144, s)-Nets in Base 3

Best Known (144−19, 144, s)-Nets in Base 3

(144−19, 144, 19691)-Net over F₃ — Constructive and digital

Digital (125, 144, 19691)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (2, 11, 8)-net over F₃, using
  - net from sequence [i] based on digital (2, 7)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 2 and N(F) ≥ 8, using
      - an explicitly constructive algebraic function field [i]
2. digital (114, 133, 19683)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3¹³³, 19683, F₃, 19, 19) (dual of [(19683, 19), 373844, 20]-NRT-code), using
    - OOA 9-folding and stacking with additional row [i] based on 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]

(144−19, 144, 59067)-Net over F₃ — Digital

Digital (125, 144, 59067)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹⁴⁴, 59067, F₃, 3, 19) (dual of [(59067, 3), 177057, 20]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3¹⁴⁴, 177201, F₃, 19) (dual of [177201, 177057, 20]-code), using
  - discarding factors / shortening the dual code based on linear OA(3¹⁴⁴, 177203, F₃, 19) (dual of [177203, 177059, 20]-code), using
    - constructionÂ X applied to C([0,9]) âŠ‚ C([0,6]) [i] based on
      1. 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]
      2. linear OA(3⁸⁹, 177148, F₃, 13) (dual of [177148, 177059, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 177148Â | 3²²âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]
      3. linear OA(3¹¹, 55, F₃, 5) (dual of [55, 44, 6]-code), using
        discarding factors / shortening the dual code based on linear OA(3¹¹, 85, F₃, 5) (dual of [85, 74, 6]-code), using
        codes constructed by inverting constructionÂ Y₁ [i]

(144−19, 144, large)-Net in Base 3 — Upper bound on s

There is no (125, 144, large)-net in base 3, because

17 times m-reduction [i] would yield (125, 127, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]