MinT - Best Known (218−31, 218, s)-Nets in Base 3

Best Known (218−31, 218, s)-Nets in Base 3

(218−31, 218, 3944)-Net over F₃ — Constructive and digital

Digital (187, 218, 3944)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (2, 17, 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 (170, 201, 3936)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3²⁰¹, 3936, F₃, 31, 31) (dual of [(3936, 31), 121815, 32]-NRT-code), using
    - OOA 15-folding and stacking with additional row [i] based on linear OA(3²⁰¹, 59041, F₃, 31) (dual of [59041, 58840, 32]-code), using
      - discarding factors / shortening the dual code based on linear OA(3²⁰¹, 59049, F₃, 31) (dual of [59049, 58848, 32]-code), using
        an extension C_e(30) of the primitive narrow-sense BCH-code C(I) with length 59048Â = 3¹⁰âˆ’1, defining interval IÂ =Â [1,30], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 31 [i]

(218−31, 218, 27052)-Net over F₃ — Digital

Digital (187, 218, 27052)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3²¹⁸, 27052, F₃, 2, 31) (dual of [(27052, 2), 53886, 32]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3²¹⁸, 29558, F₃, 2, 31) (dual of [(29558, 2), 58898, 32]-NRT-code), using
  - OOA 2-folding [i] based on linear OA(3²¹⁸, 59116, F₃, 31) (dual of [59116, 58898, 32]-code), using
    - constructionÂ X applied to C_e(30) âŠ‚ C_e(22) [i] based on
      1. linear OA(3²⁰¹, 59049, F₃, 31) (dual of [59049, 58848, 32]-code), using an extension C_e(30) of the primitive narrow-sense BCH-code C(I) with length 59048Â = 3¹⁰âˆ’1, defining interval IÂ =Â [1,30], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 31 [i]
      2. linear OA(3¹⁵¹, 59049, F₃, 23) (dual of [59049, 58898, 24]-code), using an extension C_e(22) of the primitive narrow-sense BCH-code C(I) with length 59048Â = 3¹⁰âˆ’1, defining interval IÂ =Â [1,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]
      3. linear OA(3¹⁷, 67, F₃, 7) (dual of [67, 50, 8]-code), using
        discarding factors / shortening the dual code based on linear OA(3¹⁷, 80, F₃, 7) (dual of [80, 63, 8]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 80Â = 3⁴âˆ’1, defining interval IÂ =Â [0,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]

(218−31, 218, large)-Net in Base 3 — Upper bound on s

There is no (187, 218, large)-net in base 3, because

29 times m-reduction [i] would yield (187, 189, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]