MinT - Best Known (82, 82+22, s)-Nets in Base 3

Best Known (82, 82+22, s)-Nets in Base 3

(82, 82+22, 600)-Net over F₃ — Constructive and digital

Digital (82, 104, 600)-net over F₃, using

trace code for nets [i] based on digital (4, 26, 150)-net over F₈₁, using
- net from sequence [i] based on digital (4, 149)-sequence over F₈₁, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 4 and N(F) ≥ 150, using
    - a function field by SÃ©mirat [i]

(82, 82+22, 1171)-Net over F₃ — Digital

Digital (82, 104, 1171)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁰⁴, 1171, F₃, 22) (dual of [1171, 1067, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹⁰⁴, 2206, F₃, 22) (dual of [2206, 2102, 23]-code), using
  - 1 times code embedding in larger space [i] based on linear OA(3¹⁰³, 2205, F₃, 22) (dual of [2205, 2102, 23]-code), using
    - constructionÂ X applied to C_e(21) âŠ‚ C_e(18) [i] based on
      1. linear OA(3⁹⁹, 2187, F₃, 22) (dual of [2187, 2088, 23]-code), using an extension C_e(21) of the primitive narrow-sense BCH-code C(I) with length 2186Â = 3⁷âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [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(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]

(82, 82+22, 79596)-Net in Base 3 — Upper bound on s

There is no (82, 104, 79597)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 41 751111 337001 682684 855178 987711 530412 491390 151275 > 3¹⁰⁴ [i]