MinT - Best Known (209−21, 209, s)-Nets in Base 3

Best Known (209−21, 209, s)-Nets in Base 3

(209−21, 209, 478306)-Net over F₃ — Constructive and digital

Digital (188, 209, 478306)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (3, 13, 10)-net over F₃, using
  - net from sequence [i] based on digital (3, 9)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 3 and N(F) ≥ 10, using
      - an explicitly constructive algebraic function field [i]
2. digital (175, 196, 478296)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3¹⁹⁶, 478296, F₃, 21, 21) (dual of [(478296, 21), 10044020, 22]-NRT-code), using
    - OOA 10-folding and stacking with additional row [i] based on linear OA(3¹⁹⁶, 4782961, F₃, 21) (dual of [4782961, 4782765, 22]-code), using
      - discarding factors / shortening the dual code based on linear OA(3¹⁹⁶, 4782968, F₃, 21) (dual of [4782968, 4782772, 22]-code), using
        1 times truncation [i] based on linear OA(3¹⁹⁷, 4782969, F₃, 22) (dual of [4782969, 4782772, 23]-code), using
        an extension C_e(21) of the primitive narrow-sense BCH-code C(I) with length 4782968Â = 3¹⁴âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]

(209−21, 209, 1594346)-Net over F₃ — Digital

Digital (188, 209, 1594346)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3²⁰⁹, 1594346, F₃, 3, 21) (dual of [(1594346, 3), 4782829, 22]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3²⁰⁹, 4783038, F₃, 21) (dual of [4783038, 4782829, 22]-code), using
  - 1 times code embedding in larger space [i] based on linear OA(3²⁰⁸, 4783037, F₃, 21) (dual of [4783037, 4782829, 22]-code), using
    - constructionÂ X applied to C([0,10]) âŠ‚ C([0,7]) [i] based on
      1. linear OA(3¹⁹⁷, 4782970, F₃, 21) (dual of [4782970, 4782773, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 4782970Â | 3²⁸âˆ’1, defining interval IÂ =Â [0,10], and minimum distance dÂ â‰¥ |{−10,−9,…,10}|+1Â =Â 22 (BCH-bound) [i]
      2. linear OA(3¹⁴¹, 4782970, F₃, 15) (dual of [4782970, 4782829, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 4782970Â | 3²⁸âˆ’1, defining interval IÂ =Â [0,7], and minimum distance dÂ â‰¥ |{−7,−6,…,7}|+1Â =Â 16 (BCH-bound) [i]
      3. linear OA(3¹¹, 67, F₃, 5) (dual of [67, 56, 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]

(209−21, 209, large)-Net in Base 3 — Upper bound on s

There is no (188, 209, large)-net in base 3, because

19 times m-reduction [i] would yield (188, 190, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]