MinT - Best Known (116−20, 116, s)-Nets in Base 3

Best Known (116−20, 116, s)-Nets in Base 3

(116−20, 116, 688)-Net over F₃ — Constructive and digital

Digital (96, 116, 688)-net over F₃, using

t-expansion [i] based on digital (94, 116, 688)-net over F₃, using
- trace code for nets [i] based on digital (7, 29, 172)-net over F₈₁, using
  - net from sequence [i] based on digital (7, 171)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 7 and N(F) ≥ 172, using
      - a function field by SÃ©mirat [i]

(116−20, 116, 4204)-Net over F₃ — Digital

Digital (96, 116, 4204)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹¹⁶, 4204, F₃, 20) (dual of [4204, 4088, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹¹⁶, 6602, F₃, 20) (dual of [6602, 6486, 21]-code), using
  - constructionÂ X applied to C_e(19) âŠ‚ C_e(13) [i] based on
    1. linear OA(3¹⁰⁵, 6561, F₃, 20) (dual of [6561, 6456, 21]-code), using an extension C_e(19) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 3⁸âˆ’1, defining interval IÂ =Â [1,19], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [i]
    2. linear OA(3⁷³, 6561, F₃, 14) (dual of [6561, 6488, 15]-code), using an extension C_e(13) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 3⁸âˆ’1, defining interval IÂ =Â [1,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [i]
    3. linear OA(3¹¹, 41, F₃, 5) (dual of [41, 30, 6]-code), using
      - (u,Â u+v)-construction [i] based on
        linear OA(3³, 13, F₃, 2) (dual of [13, 10, 3]-code), using
        Hamming code H(3,3) [i]
        linear OA(3⁸, 28, F₃, 5) (dual of [28, 20, 6]-code), using
        dual code (with bound on d by constructionÂ Y1) [i] based on
        linear OA(3²⁰, 28, F₃, 14) (dual of [28, 8, 15]-code), using
        contraction [i] based on linear OA(3⁴⁸, 56, F₃, 29) (dual of [56, 8, 30]-code), using the BCH-code C(I) with length 56Â | 3⁶âˆ’1, defining interval IÂ =Â {0,1,…,28}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 30 [i]
        nonexistence of linear OA(3¹⁹, 23, F₃, 14) (dual of [23, 4, 15]-code), because
        2 times truncation [i] would yield linear OA(3¹⁷, 21, F₃, 12) (dual of [21, 4, 13]-code), but
        â€œHNâ€ bound on codes from Brouwerâ€™s database [i]

(116−20, 116, 775438)-Net in Base 3 — Upper bound on s

There is no (96, 116, 775439)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 22 185362 320282 891833 898390 503608 163577 130801 768542 142477 > 3¹¹⁶ [i]

Best Known (116−20, 116, s)-Nets in Base 3

(116−20, 116, 688)-Net over F3 — Constructive and digital

(116−20, 116, 4204)-Net over F3 — Digital

(116−20, 116, 775438)-Net in Base 3 — Upper bound on s

(116−20, 116, 688)-Net over F₃ — Constructive and digital

(116−20, 116, 4204)-Net over F₃ — Digital