MinT - Best Known (181−38, 181, s)-Nets in Base 3

Best Known (181−38, 181, s)-Nets in Base 3

(181−38, 181, 688)-Net over F₃ — Constructive and digital

Digital (143, 181, 688)-net over F₃, using

3¹ times duplication [i] based on digital (142, 180, 688)-net over F₃, using
- trace code for nets [i] based on digital (7, 45, 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]

(181−38, 181, 1702)-Net over F₃ — Digital

Digital (143, 181, 1702)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁸¹, 1702, F₃, 38) (dual of [1702, 1521, 39]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹⁸¹, 2207, F₃, 38) (dual of [2207, 2026, 39]-code), using
  - constructionÂ X applied to C_e(37) âŠ‚ C_e(33) [i] based on
    1. linear OA(3¹⁷⁶, 2187, F₃, 38) (dual of [2187, 2011, 39]-code), using an extension C_e(37) of the primitive narrow-sense BCH-code C(I) with length 2186Â = 3⁷âˆ’1, defining interval IÂ =Â [1,37], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 38 [i]
    2. linear OA(3¹⁵⁵, 2187, F₃, 34) (dual of [2187, 2032, 35]-code), using an extension C_e(33) of the primitive narrow-sense BCH-code C(I) with length 2186Â = 3⁷âˆ’1, defining interval IÂ =Â [1,33], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 34 [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]

(181−38, 181, 139102)-Net in Base 3 — Upper bound on s

There is no (143, 181, 139103)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 228 541995 426278 889922 518137 618442 379207 181062 236214 986407 405006 530726 922861 157642 702683 > 3¹⁸¹ [i]