MinT - Best Known (206−25, 206, s)-Nets in Base 3

Best Known (206−25, 206, s)-Nets in Base 3

(206−25, 206, 44293)-Net over F₃ — Constructive and digital

Digital (181, 206, 44293)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (1, 13, 7)-net over F₃, using
  - net from sequence [i] based on digital (1, 6)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 1 and N(F) ≥ 7, using
      - an explicitly constructive algebraic function field [i]
2. digital (168, 193, 44286)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3¹⁹³, 44286, F₃, 25, 25) (dual of [(44286, 25), 1106957, 26]-NRT-code), using
    - OOA 12-folding and stacking with additional row [i] based on linear OA(3¹⁹³, 531433, F₃, 25) (dual of [531433, 531240, 26]-code), using
      - discarding factors / shortening the dual code based on linear OA(3¹⁹³, 531441, F₃, 25) (dual of [531441, 531248, 26]-code), using
        an extension C_e(24) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 3¹²âˆ’1, defining interval IÂ =Â [1,24], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 25 [i]

(206−25, 206, 177167)-Net over F₃ — Digital

Digital (181, 206, 177167)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3²⁰⁶, 177167, F₃, 3, 25) (dual of [(177167, 3), 531295, 26]-NRT-code), using
- 3² times duplication [i] based on linear OOA(3²⁰⁴, 177167, F₃, 3, 25) (dual of [(177167, 3), 531297, 26]-NRT-code), using
  - OOA 3-folding [i] based on linear OA(3²⁰⁴, 531501, F₃, 25) (dual of [531501, 531297, 26]-code), using
    - constructionÂ X applied to C([0,12]) âŠ‚ C([0,9]) [i] based on
      1. linear OA(3¹⁹³, 531442, F₃, 25) (dual of [531442, 531249, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442Â | 3²⁴âˆ’1, defining interval IÂ =Â [0,12], and minimum distance dÂ â‰¥ |{−12,−11,…,12}|+1Â =Â 26 (BCH-bound) [i]
      2. 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]
      3. 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]

(206−25, 206, large)-Net in Base 3 — Upper bound on s

There is no (181, 206, large)-net in base 3, because

23 times m-reduction [i] would yield (181, 183, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]