MinT - Best Known (213−37, 213, s)-Nets in Base 3

Best Known (213−37, 213, s)-Nets in Base 3

(213−37, 213, 1480)-Net over F₃ — Constructive and digital

Digital (176, 213, 1480)-net over F₃, using

3¹ times duplication [i] based on digital (175, 212, 1480)-net over F₃, using
- trace code for nets [i] based on digital (16, 53, 370)-net over F₈₁, using
  - net from sequence [i] based on digital (16, 369)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 16 and N(F) ≥ 370, using
      - a function field by GarcÃa and Quoos [i]

(213−37, 213, 5364)-Net over F₃ — Digital

Digital (176, 213, 5364)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3²¹³, 5364, F₃, 37) (dual of [5364, 5151, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(3²¹³, 6621, F₃, 37) (dual of [6621, 6408, 38]-code), using
  - 4 times code embedding in larger space [i] based on linear OA(3²⁰⁹, 6617, F₃, 37) (dual of [6617, 6408, 38]-code), using
    - constructionÂ X applied to C_e(36) âŠ‚ C_e(28) [i] based on
      1. 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]
      2. linear OA(3¹⁵³, 6561, F₃, 29) (dual of [6561, 6408, 30]-code), using an extension C_e(28) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 3⁸âˆ’1, defining interval IÂ =Â [1,28], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [i]
      3. linear OA(3¹⁶, 56, F₃, 7) (dual of [56, 40, 8]-code), using
        discarding factors / shortening the dual code based on linear OA(3¹⁶, 82, F₃, 7) (dual of [82, 66, 8]-code), using
        generator matrices provided on Edelâ€™s homepage [i]

(213−37, 213, 1572256)-Net in Base 3 — Upper bound on s

There is no (176, 213, 1572257)-net in base 3, because

1 times m-reduction [i] would yield (176, 212, 1572257)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 141159 326210 619335 465775 227868 969242 437939 632737 053492 818818 233414 420260 496277 531240 361230 448224 355721 > 3²¹² [i]