MinT - Best Known (132−28, 132, s)-Nets in Base 3

Best Known (132−28, 132, s)-Nets in Base 3

(132−28, 132, 640)-Net over F₃ — Constructive and digital

Digital (104, 132, 640)-net over F₃, using

trace code for nets [i] based on digital (5, 33, 160)-net over F₈₁, using
- net from sequence [i] based on digital (5, 159)-sequence over F₈₁, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 5 and N(F) ≥ 160, using
    - a function field by SÃ©mirat [i]

(132−28, 132, 1313)-Net over F₃ — Digital

Digital (104, 132, 1313)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹³², 1313, F₃, 28) (dual of [1313, 1181, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹³², 2206, F₃, 28) (dual of [2206, 2074, 29]-code), using
  - 1 times code embedding in larger space [i] based on linear OA(3¹³¹, 2205, F₃, 28) (dual of [2205, 2074, 29]-code), using
    - constructionÂ X applied to C_e(27) âŠ‚ C_e(24) [i] based on
      1. linear OA(3¹²⁷, 2187, F₃, 28) (dual of [2187, 2060, 29]-code), using an extension C_e(27) of the primitive narrow-sense BCH-code C(I) with length 2186Â = 3⁷âˆ’1, defining interval IÂ =Â [1,27], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 28 [i]
      2. linear OA(3¹¹³, 2187, F₃, 25) (dual of [2187, 2074, 26]-code), using an extension C_e(24) of the primitive narrow-sense BCH-code C(I) with length 2186Â = 3⁷âˆ’1, defining interval IÂ =Â [1,24], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 25 [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]

(132−28, 132, 95266)-Net in Base 3 — Upper bound on s

There is no (104, 132, 95267)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 955 141953 245761 563148 631006 473697 787627 719997 862474 205758 649229 > 3¹³² [i]