MinT - Best Known (131−18, 131, s)-Nets in Base 3

Best Known (131−18, 131, s)-Nets in Base 3

(131−18, 131, 6569)-Net over F₃ — Constructive and digital

Digital (113, 131, 6569)-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 (102, 120, 6561)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3¹²⁰, 6561, F₃, 18, 18) (dual of [(6561, 18), 117978, 19]-NRT-code), using
    - OA 9-folding and stacking [i] based on linear OA(3¹²⁰, 59049, F₃, 18) (dual of [59049, 58929, 19]-code), using
      - 1 times truncation [i] based on linear OA(3¹²¹, 59050, F₃, 19) (dual of [59050, 58929, 20]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 59050Â | 3²⁰âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (BCH-bound) [i]

(131−18, 131, 29550)-Net over F₃ — Digital

Digital (113, 131, 29550)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹³¹, 29550, F₃, 2, 18) (dual of [(29550, 2), 58969, 19]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3¹³¹, 59100, F₃, 18) (dual of [59100, 58969, 19]-code), using
  - 1 times truncation [i] based on linear OA(3¹³², 59101, F₃, 19) (dual of [59101, 58969, 20]-code), using
    - constructionÂ X applied to C([0,9]) âŠ‚ C([0,6]) [i] based on
      1. linear OA(3¹²¹, 59050, F₃, 19) (dual of [59050, 58929, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 59050Â | 3²⁰âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (BCH-bound) [i]
      2. linear OA(3⁸¹, 59050, F₃, 13) (dual of [59050, 58969, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 59050Â | 3²⁰âˆ’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, 51, F₃, 5) (dual of [51, 40, 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]

(131−18, 131, large)-Net in Base 3 — Upper bound on s

There is no (113, 131, large)-net in base 3, because

16 times m-reduction [i] would yield (113, 115, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]