MinT - Best Known (228−48, 228, s)-Nets in Base 3

Best Known (228−48, 228, s)-Nets in Base 3

(228−48, 228, 688)-Net over F₃ — Constructive and digital

Digital (180, 228, 688)-net over F₃, using

t-expansion [i] based on digital (178, 228, 688)-net over F₃, using
- trace code for nets [i] based on digital (7, 57, 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]

(228−48, 228, 1993)-Net over F₃ — Digital

Digital (180, 228, 1993)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3²²⁸, 1993, F₃, 48) (dual of [1993, 1765, 49]-code), using
- discarding factors / shortening the dual code based on linear OA(3²²⁸, 2201, F₃, 48) (dual of [2201, 1973, 49]-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(3³, 13, F₃, 2) (dual of [13, 10, 3]-code), using
      - Hamming code H(3,3) [i]

(228−48, 228, 167079)-Net in Base 3 — Upper bound on s

There is no (180, 228, 167080)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 6 076692 378275 674701 940084 540595 036328 409170 435731 476725 675773 257218 795671 414881 189565 781107 839484 565549 251841 > 3²²⁸ [i]