MinT - Best Known (208−44, 208, s)-Nets in Base 3

Best Known (208−44, 208, s)-Nets in Base 3

(208−44, 208, 688)-Net over F₃ — Constructive and digital

Digital (164, 208, 688)-net over F₃, using

t-expansion [i] based on digital (163, 208, 688)-net over F₃, using
- trace code for nets [i] based on digital (7, 52, 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]

(208−44, 208, 1816)-Net over F₃ — Digital

Digital (164, 208, 1816)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3²⁰⁸, 1816, F₃, 44) (dual of [1816, 1608, 45]-code), using
- discarding factors / shortening the dual code based on linear OA(3²⁰⁸, 2205, F₃, 44) (dual of [2205, 1997, 45]-code), using
  - constructionÂ X applied to C_e(43) âŠ‚ C_e(40) [i] based on
    1. linear OA(3²⁰⁴, 2187, F₃, 44) (dual of [2187, 1983, 45]-code), using an extension C_e(43) of the primitive narrow-sense BCH-code C(I) with length 2186Â = 3⁷âˆ’1, defining interval IÂ =Â [1,43], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 44 [i]
    2. linear OA(3¹⁹⁰, 2187, F₃, 41) (dual of [2187, 1997, 42]-code), using an extension C_e(40) of the primitive narrow-sense BCH-code C(I) with length 2186Â = 3⁷âˆ’1, defining interval IÂ =Â [1,40], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 41 [i]
    3. linear OA(3⁴, 18, F₃, 2) (dual of [18, 14, 3]-code), using
      - discarding factors / shortening the dual code based on linear OA(3⁴, 40, F₃, 2) (dual of [40, 36, 3]-code), using
        Hamming code H(4,3) [i]

(208−44, 208, 146799)-Net in Base 3 — Upper bound on s

There is no (164, 208, 146800)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1742 872728 509521 041132 847257 550476 733353 523253 843563 957028 551969 173745 188984 474562 104770 074671 416481 > 3²⁰⁸ [i]