MinT - Best Known (213−40, 213, s)-Nets in Base 3

Best Known (213−40, 213, s)-Nets in Base 3

(213−40, 213, 896)-Net over F₃ — Constructive and digital

Digital (173, 213, 896)-net over F₃, using

3¹ times duplication [i] based on digital (172, 212, 896)-net over F₃, using
- trace code for nets [i] based on digital (13, 53, 224)-net over F₈₁, using
  - net from sequence [i] based on digital (13, 223)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 13 and N(F) ≥ 224, using
      - a function field by SÃ©mirat [i]

(213−40, 213, 3412)-Net over F₃ — Digital

Digital (173, 213, 3412)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3²¹³, 3412, F₃, 40) (dual of [3412, 3199, 41]-code), using
- discarding factors / shortening the dual code based on linear OA(3²¹³, 6581, F₃, 40) (dual of [6581, 6368, 41]-code), using
  - constructionÂ X applied to C_e(39) âŠ‚ C_e(36) [i] based on
    1. linear OA(3²⁰⁹, 6561, F₃, 40) (dual of [6561, 6352, 41]-code), using an extension C_e(39) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 3⁸âˆ’1, defining interval IÂ =Â [1,39], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 40 [i]
    2. linear OA(3¹⁹³, 6561, F₃, 37) (dual of [6561, 6368, 38]-code), using an extension C_e(36) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 3⁸âˆ’1, defining interval IÂ =Â [1,36], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 37 [i]
    3. linear OA(3⁴, 20, F₃, 2) (dual of [20, 16, 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]

(213−40, 213, 500726)-Net in Base 3 — Upper bound on s

There is no (173, 213, 500727)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 423480 814617 984900 228657 071725 182535 610840 454913 536777 240463 902821 494163 060134 224930 543879 975004 717401 > 3²¹³ [i]