MinT - Best Known (237−31, 237, s)-Nets in Base 3

Best Known (237−31, 237, s)-Nets in Base 3

(237−31, 237, 11816)-Net over F₃ — Constructive and digital

Digital (206, 237, 11816)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (1, 16, 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 (190, 221, 11809)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3²²¹, 11809, F₃, 31, 31) (dual of [(11809, 31), 365858, 32]-NRT-code), using
    - OOA 15-folding and stacking with additional row [i] based on linear OA(3²²¹, 177136, F₃, 31) (dual of [177136, 176915, 32]-code), using
      - discarding factors / shortening the dual code based on linear OA(3²²¹, 177147, F₃, 31) (dual of [177147, 176926, 32]-code), using
        an extension C_e(30) of the primitive narrow-sense BCH-code C(I) with length 177146Â = 3¹¹âˆ’1, defining interval IÂ =Â [1,30], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 31 [i]

(237−31, 237, 59072)-Net over F₃ — Digital

Digital (206, 237, 59072)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3²³⁷, 59072, F₃, 3, 31) (dual of [(59072, 3), 176979, 32]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3²³⁷, 177216, F₃, 31) (dual of [177216, 176979, 32]-code), using
  - discarding factors / shortening the dual code based on linear OA(3²³⁷, 177218, F₃, 31) (dual of [177218, 176981, 32]-code), using
    - constructionÂ X applied to C_e(30) âŠ‚ C_e(22) [i] based on
      1. linear OA(3²²¹, 177147, F₃, 31) (dual of [177147, 176926, 32]-code), using an extension C_e(30) of the primitive narrow-sense BCH-code C(I) with length 177146Â = 3¹¹âˆ’1, defining interval IÂ =Â [1,30], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 31 [i]
      2. linear OA(3¹⁶⁶, 177147, F₃, 23) (dual of [177147, 176981, 24]-code), using an extension C_e(22) of the primitive narrow-sense BCH-code C(I) with length 177146Â = 3¹¹âˆ’1, defining interval IÂ =Â [1,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]
      3. linear OA(3¹⁶, 71, F₃, 7) (dual of [71, 55, 8]-code), using
        discarding factors / shortening the dual code based on linear OA(3¹⁶, 82, F₃, 7) (dual of [82, 66, 8]-code), using
        generator matrices provided on Edelâ€™s homepage [i]

(237−31, 237, large)-Net in Base 3 — Upper bound on s

There is no (206, 237, large)-net in base 3, because

29 times m-reduction [i] would yield (206, 208, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]