MinT - Best Known (140−16, 140, s)-Nets in Base 3

Best Known (140−16, 140, s)-Nets in Base 3

(140−16, 140, 199297)-Net over F₃ — Constructive and digital

Digital (124, 140, 199297)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (1, 9, 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 (115, 131, 199290)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3¹³¹, 199290, F₃, 16, 16) (dual of [(199290, 16), 3188509, 17]-NRT-code), using
    - OA 8-folding and stacking [i] based on linear OA(3¹³¹, 1594320, F₃, 16) (dual of [1594320, 1594189, 17]-code), using
      - discarding factors / shortening the dual code based on linear OA(3¹³¹, 1594323, F₃, 16) (dual of [1594323, 1594192, 17]-code), using
        an extension C_e(15) of the primitive narrow-sense BCH-code C(I) with length 1594322Â = 3¹³âˆ’1, defining interval IÂ =Â [1,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]

(140−16, 140, 531457)-Net over F₃ — Digital

Digital (124, 140, 531457)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹⁴⁰, 531457, F₃, 3, 16) (dual of [(531457, 3), 1594231, 17]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3¹⁴⁰, 1594371, F₃, 16) (dual of [1594371, 1594231, 17]-code), using
  - constructionÂ X applied to C_e(15) âŠ‚ C_e(10) [i] based on
    1. linear OA(3¹³¹, 1594323, F₃, 16) (dual of [1594323, 1594192, 17]-code), using an extension C_e(15) of the primitive narrow-sense BCH-code C(I) with length 1594322Â = 3¹³âˆ’1, defining interval IÂ =Â [1,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]
    2. linear OA(3⁹², 1594323, F₃, 11) (dual of [1594323, 1594231, 12]-code), using an extension C_e(10) of the primitive narrow-sense BCH-code C(I) with length 1594322Â = 3¹³âˆ’1, defining interval IÂ =Â [1,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]
    3. linear OA(3⁹, 48, F₃, 4) (dual of [48, 39, 5]-code), using
      - discarding factors / shortening the dual code based on linear OA(3⁹, 80, F₃, 4) (dual of [80, 71, 5]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 80Â = 3⁴âˆ’1, defining interval IÂ =Â [0,2], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]

(140−16, 140, large)-Net in Base 3 — Upper bound on s

There is no (124, 140, large)-net in base 3, because

14 times m-reduction [i] would yield (124, 126, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]