MinT - Best Known (164−30, 164, s)-Nets in Base 3

Best Known (164−30, 164, s)-Nets in Base 3

(164−30, 164, 695)-Net over F₃ — Constructive and digital

Digital (134, 164, 695)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (1, 16, 7)-net over F₃, using
  - net from sequence [i] based on digital (1, 6)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 1 and N(F) ≥ 7, using
      - an explicitly constructive algebraic function field [i]
2. digital (118, 148, 688)-net over F₃, using
  - trace code for nets [i] based on digital (7, 37, 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]

(164−30, 164, 3358)-Net over F₃ — Digital

Digital (134, 164, 3358)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁶⁴, 3358, F₃, 30) (dual of [3358, 3194, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹⁶⁴, 6575, F₃, 30) (dual of [6575, 6411, 31]-code), using
  - constructionÂ X applied to C([0,15]) âŠ‚ C([0,13]) [i] based on
    1. linear OA(3¹⁶¹, 6562, F₃, 31) (dual of [6562, 6401, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562Â | 3¹⁶âˆ’1, defining interval IÂ =Â [0,15], and minimum distance dÂ â‰¥ |{−15,−14,…,15}|+1Â =Â 32 (BCH-bound) [i]
    2. linear OA(3¹⁴⁵, 6562, F₃, 27) (dual of [6562, 6417, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562Â | 3¹⁶âˆ’1, defining interval IÂ =Â [0,13], and minimum distance dÂ â‰¥ |{−13,−12,…,13}|+1Â =Â 28 (BCH-bound) [i]
    3. linear OA(3³, 13, F₃, 2) (dual of [13, 10, 3]-code), using
      - Hamming code H(3,3) [i]

(164−30, 164, 528750)-Net in Base 3 — Upper bound on s

There is no (134, 164, 528751)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1 769689 161697 926277 541164 795953 604789 747186 790447 667297 983499 927794 688792 767907 > 3¹⁶⁴ [i]