MinT - Best Known (182, 231, s)-Nets in Base 3

Best Known (182, 231, s)-Nets in Base 3

(182, 231, 688)-Net over F₃ — Constructive and digital

Digital (182, 231, 688)-net over F₃, using

t-expansion [i] based on digital (181, 231, 688)-net over F₃, using
- 1 times m-reduction [i] based on digital (181, 232, 688)-net over F₃, using
  - trace code for nets [i] based on digital (7, 58, 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]

(182, 231, 1942)-Net over F₃ — Digital

Digital (182, 231, 1942)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3²³¹, 1942, F₃, 49) (dual of [1942, 1711, 50]-code), using
- discarding factors / shortening the dual code based on linear OA(3²³¹, 2208, F₃, 49) (dual of [2208, 1977, 50]-code), using
  - 1 times code embedding in larger space [i] based on linear OA(3²³⁰, 2207, F₃, 49) (dual of [2207, 1977, 50]-code), using
    - constructionÂ X applied to C([0,24]) âŠ‚ C([0,22]) [i] based on
      1. linear OA(3²²⁵, 2188, F₃, 49) (dual of [2188, 1963, 50]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188Â | 3¹⁴âˆ’1, defining interval IÂ =Â [0,24], and minimum distance dÂ â‰¥ |{−24,−23,…,24}|+1Â =Â 50 (BCH-bound) [i]
      2. linear OA(3²¹¹, 2188, F₃, 45) (dual of [2188, 1977, 46]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188Â | 3¹⁴âˆ’1, defining interval IÂ =Â [0,22], and minimum distance dÂ â‰¥ |{−22,−21,…,22}|+1Â =Â 46 (BCH-bound) [i]
      3. linear OA(3⁵, 19, F₃, 3) (dual of [19, 14, 4]-code or 19-cap in PG(4,3)), using
        discarding factors / shortening the dual code based on 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]

(182, 231, 183100)-Net in Base 3 — Upper bound on s

There is no (182, 231, 183101)-net in base 3, because

1 times m-reduction [i] would yield (182, 230, 183101)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 54 692654 187282 671095 667790 133787 851819 828164 662925 248987 098625 519581 562025 975797 825024 446101 204095 477053 556001 > 3²³⁰ [i]