MinT - Best Known (108, 108+26, s)-Nets in Base 3

Best Known (108, 108+26, s)-Nets in Base 3

(108, 108+26, 688)-Net over F₃ — Constructive and digital

Digital (108, 134, 688)-net over F₃, using

3² times duplication [i] based on digital (106, 132, 688)-net over F₃, using
- trace code for nets [i] based on digital (7, 33, 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]

(108, 108+26, 2137)-Net over F₃ — Digital

Digital (108, 134, 2137)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹³⁴, 2137, F₃, 26) (dual of [2137, 2003, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹³⁴, 2236, F₃, 26) (dual of [2236, 2102, 27]-code), using
  - constructionÂ X applied to C_e(25) âŠ‚ C_e(18) [i] based on
    1. linear OA(3¹²⁰, 2187, F₃, 26) (dual of [2187, 2067, 27]-code), using an extension C_e(25) of the primitive narrow-sense BCH-code C(I) with length 2186Â = 3⁷âˆ’1, defining interval IÂ =Â [1,25], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 26 [i]
    2. linear OA(3⁸⁵, 2187, F₃, 19) (dual of [2187, 2102, 20]-code), using an extension C_e(18) of the primitive narrow-sense BCH-code C(I) with length 2186Â = 3⁷âˆ’1, defining interval IÂ =Â [1,18], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]
    3. linear OA(3¹⁴, 49, F₃, 6) (dual of [49, 35, 7]-code), using
      - discarding factors / shortening the dual code based on linear OA(3¹⁴, 53, F₃, 6) (dual of [53, 39, 7]-code), using
        codes constructed by inverting constructionÂ Y₁ [i]

(108, 108+26, 234623)-Net in Base 3 — Upper bound on s

There is no (108, 134, 234624)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 8595 122163 063991 648534 757226 221779 503816 240989 063620 450950 996225 > 3¹³⁴ [i]