MinT - Best Known (153−32, 153, s)-Nets in Base 3

Best Known (153−32, 153, s)-Nets in Base 3

(153−32, 153, 640)-Net over F₃ — Constructive and digital

Digital (121, 153, 640)-net over F₃, using

3¹ times duplication [i] based on digital (120, 152, 640)-net over F₃, using
- t-expansion [i] based on digital (119, 152, 640)-net over F₃, using
  - trace code for nets [i] based on digital (5, 38, 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]

(153−32, 153, 1547)-Net over F₃ — Digital

Digital (121, 153, 1547)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁵³, 1547, F₃, 32) (dual of [1547, 1394, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹⁵³, 2207, F₃, 32) (dual of [2207, 2054, 33]-code), using
  - constructionÂ X applied to C_e(31) âŠ‚ C_e(27) [i] based on
    1. linear OA(3¹⁴⁸, 2187, F₃, 32) (dual of [2187, 2039, 33]-code), using an extension C_e(31) of the primitive narrow-sense BCH-code C(I) with length 2186Â = 3⁷âˆ’1, defining interval IÂ =Â [1,31], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 32 [i]
    2. 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]
    3. linear OA(3⁵, 20, F₃, 3) (dual of [20, 15, 4]-code or 20-cap in PG(4,3)), using
      - doubling the cap [i] based on linear OA(3⁴, 10, F₃, 3) (dual of [10, 6, 4]-code or 10-cap in PG(3,3)), using
        ovoid in PG(3,Â 3) [i]

(153−32, 153, 124144)-Net in Base 3 — Upper bound on s

There is no (121, 153, 124145)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 9 990691 229498 266689 934136 477874 050173 780139 562766 459859 821953 285243 145025 > 3¹⁵³ [i]