MinT - Best Known (123, 153, s)-Nets in Base 3

Best Known (123, 153, s)-Nets in Base 3

(123, 153, 688)-Net over F₃ — Constructive and digital

Digital (123, 153, 688)-net over F₃, using

3¹ times duplication [i] based on digital (122, 152, 688)-net over F₃, using
- t-expansion [i] based on digital (121, 152, 688)-net over F₃, using
  - trace code for nets [i] based on digital (7, 38, 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]

(123, 153, 2172)-Net over F₃ — Digital

Digital (123, 153, 2172)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁵³, 2172, F₃, 30) (dual of [2172, 2019, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹⁵³, 2228, F₃, 30) (dual of [2228, 2075, 31]-code), using
  - 4 times code embedding in larger space [i] based on linear OA(3¹⁴⁹, 2224, F₃, 30) (dual of [2224, 2075, 31]-code), using
    - constructionÂ X applied to C([0,15]) âŠ‚ C([0,12]) [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(3¹¹³, 2188, F₃, 25) (dual of [2188, 2075, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188Â | 3¹⁴âˆ’1, defining interval IÂ =Â [0,12], and minimum distance dÂ â‰¥ |{−12,−11,…,12}|+1Â =Â 26 (BCH-bound) [i]
      3. linear OA(3⁸, 36, F₃, 4) (dual of [36, 28, 5]-code), using
        discarding factors / shortening the dual code based on linear OA(3⁸, 41, F₃, 4) (dual of [41, 33, 5]-code), using
        the narrow-sense BCH-code C(I) with length 41Â | 3⁸âˆ’1, defining interval IÂ =Â [1,1], and minimum distance dÂ â‰¥ |{−3,−1,1,3}|+1Â =Â 5 (BCH-bound) [i]

(123, 153, 236236)-Net in Base 3 — Upper bound on s

There is no (123, 153, 236237)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 9 990079 228886 273066 689508 584362 082633 479326 482306 857228 031472 231976 484251 > 3¹⁵³ [i]