MinT - Best Known (139−25, 139, s)-Nets in Base 3

Best Known (139−25, 139, s)-Nets in Base 3

(139−25, 139, 688)-Net over F₃ — Constructive and digital

Digital (114, 139, 688)-net over F₃, using

t-expansion [i] based on digital (112, 139, 688)-net over F₃, using
- 1 times m-reduction [i] based on digital (112, 140, 688)-net over F₃, using
  - trace code for nets [i] based on digital (7, 35, 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]

(139−25, 139, 3415)-Net over F₃ — Digital

Digital (114, 139, 3415)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹³⁹, 3415, F₃, 25) (dual of [3415, 3276, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹³⁹, 6598, F₃, 25) (dual of [6598, 6459, 26]-code), using
  - constructionÂ X applied to C([0,12]) âŠ‚ C([0,9]) [i] based on
    1. linear OA(3¹²⁹, 6562, F₃, 25) (dual of [6562, 6433, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562Â | 3¹⁶âˆ’1, defining interval IÂ =Â [0,12], and minimum distance dÂ â‰¥ |{−12,−11,…,12}|+1Â =Â 26 (BCH-bound) [i]
    2. linear OA(3⁹⁷, 6562, F₃, 19) (dual of [6562, 6465, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562Â | 3¹⁶âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (BCH-bound) [i]
    3. linear OA(3¹⁰, 36, F₃, 5) (dual of [36, 26, 6]-code), using
      - (u,Â uâˆ’v,Â u+v+w)-construction [i] based on
        linear OA(3¹, 12, F₃, 1) (dual of [12, 11, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(3¹, s, F₃, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]
        linear OA(3³, 12, F₃, 2) (dual of [12, 9, 3]-code), using
        discarding factors / shortening the dual code based on linear OA(3³, 13, F₃, 2) (dual of [13, 10, 3]-code), using
        Hamming code H(3,3) [i]
        linear OA(3⁶, 12, F₃, 5) (dual of [12, 6, 6]-code), using
        extended Golay code G_e(3) [i]

(139−25, 139, 811371)-Net in Base 3 — Upper bound on s

There is no (114, 139, 811372)-net in base 3, because

1 times m-reduction [i] would yield (114, 138, 811372)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 696204 851096 846088 306364 413022 953295 258916 052900 719322 290285 767361 > 3¹³⁸ [i]

Best Known (139−25, 139, s)-Nets in Base 3

(139−25, 139, 688)-Net over F3 — Constructive and digital

(139−25, 139, 3415)-Net over F3 — Digital

(139−25, 139, 811371)-Net in Base 3 — Upper bound on s

(139−25, 139, 688)-Net over F₃ — Constructive and digital

(139−25, 139, 3415)-Net over F₃ — Digital