MinT - Best Known (145−31, 145, s)-Nets in Base 3

Best Known (145−31, 145, s)-Nets in Base 3

(145−31, 145, 640)-Net over F₃ — Constructive and digital

Digital (114, 145, 640)-net over F₃, using

3¹ times duplication [i] based on digital (113, 144, 640)-net over F₃, using
- trace code for nets [i] based on digital (5, 36, 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]

(145−31, 145, 1339)-Net over F₃ — Digital

Digital (114, 145, 1339)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁴⁵, 1339, F₃, 31) (dual of [1339, 1194, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹⁴⁵, 2198, F₃, 31) (dual of [2198, 2053, 32]-code), using
  - constructionÂ X applied to C([0,15]) âŠ‚ C([0,13]) [i] based on
    1. linear OA(3¹⁴¹, 2188, F₃, 31) (dual of [2188, 2047, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188Â | 3¹⁴âˆ’1, defining interval IÂ =Â [0,15], and minimum distance dÂ â‰¥ |{−15,−14,…,15}|+1Â =Â 32 (BCH-bound) [i]
    2. linear OA(3¹²⁷, 2188, F₃, 27) (dual of [2188, 2061, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188Â | 3¹⁴âˆ’1, defining interval IÂ =Â [0,13], and minimum distance dÂ â‰¥ |{−13,−12,…,13}|+1Â =Â 28 (BCH-bound) [i]
    3. 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]

(145−31, 145, 122193)-Net in Base 3 — Upper bound on s

There is no (114, 145, 122194)-net in base 3, because

1 times m-reduction [i] would yield (114, 144, 122194)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 507 541418 056819 439769 720711 464322 305576 336084 601698 360422 976266 809945 > 3¹⁴⁴ [i]