MinT - Best Known (137, 155, s)-Nets in Base 3

Best Known (137, 155, s)-Nets in Base 3

(137, 155, 59057)-Net over F₃ — Constructive and digital

Digital (137, 155, 59057)-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 (126, 144, 59049)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3¹⁴⁴, 59049, F₃, 18, 18) (dual of [(59049, 18), 1062738, 19]-NRT-code), using
    - OA 9-folding and stacking [i] based on linear OA(3¹⁴⁴, 531441, F₃, 18) (dual of [531441, 531297, 19]-code), using
      - 1 times truncation [i] based on linear OA(3¹⁴⁵, 531442, F₃, 19) (dual of [531442, 531297, 20]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 531442Â | 3²⁴âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (BCH-bound) [i]

(137, 155, 236237)-Net over F₃ — Digital

Digital (137, 155, 236237)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹⁵⁵, 236237, F₃, 2, 18) (dual of [(236237, 2), 472319, 19]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3¹⁵⁵, 265750, F₃, 2, 18) (dual of [(265750, 2), 531345, 19]-NRT-code), using
  - OOA 2-folding [i] based on linear OA(3¹⁵⁵, 531500, F₃, 18) (dual of [531500, 531345, 19]-code), using
    - 1 times truncation [i] based on linear OA(3¹⁵⁶, 531501, F₃, 19) (dual of [531501, 531345, 20]-code), using
      - constructionÂ X applied to C([0,9]) âŠ‚ C([0,6]) [i] based on
        linear OA(3¹⁴⁵, 531442, F₃, 19) (dual of [531442, 531297, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442Â | 3²⁴âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (BCH-bound) [i]
        linear OA(3⁹⁷, 531442, F₃, 13) (dual of [531442, 531345, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442Â | 3²⁴âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]
        linear OA(3¹¹, 59, F₃, 5) (dual of [59, 48, 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]

(137, 155, large)-Net in Base 3 — Upper bound on s

There is no (137, 155, large)-net in base 3, because

16 times m-reduction [i] would yield (137, 139, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]