MinT - Best Known (209−37, 209, s)-Nets in Base 3

Best Known (209−37, 209, s)-Nets in Base 3

(209−37, 209, 896)-Net over F₃ — Constructive and digital

Digital (172, 209, 896)-net over F₃, using

3 times m-reduction [i] based on digital (172, 212, 896)-net over F₃, using
- trace code for nets [i] based on digital (13, 53, 224)-net over F₈₁, using
  - net from sequence [i] based on digital (13, 223)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 13 and N(F) ≥ 224, using
      - a function field by SÃ©mirat [i]

(209−37, 209, 4728)-Net over F₃ — Digital

Digital (172, 209, 4728)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3²⁰⁹, 4728, F₃, 37) (dual of [4728, 4519, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(3²⁰⁹, 6617, F₃, 37) (dual of [6617, 6408, 38]-code), using
  - constructionÂ X applied to C_e(36) âŠ‚ C_e(28) [i] based on
    1. linear OA(3¹⁹³, 6561, F₃, 37) (dual of [6561, 6368, 38]-code), using an extension C_e(36) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 3⁸âˆ’1, defining interval IÂ =Â [1,36], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 37 [i]
    2. linear OA(3¹⁵³, 6561, F₃, 29) (dual of [6561, 6408, 30]-code), using an extension C_e(28) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 3⁸âˆ’1, defining interval IÂ =Â [1,28], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [i]
    3. linear OA(3¹⁶, 56, F₃, 7) (dual of [56, 40, 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]

(209−37, 209, 1231671)-Net in Base 3 — Upper bound on s

There is no (172, 209, 1231672)-net in base 3, because

1 times m-reduction [i] would yield (172, 208, 1231672)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1742 694278 815879 352188 774888 069978 229182 688056 148521 643907 505936 540222 402326 525677 135491 989913 104081 > 3²⁰⁸ [i]