MinT - Best Known (123−25, 123, s)-Nets in Base 3

Best Known (123−25, 123, s)-Nets in Base 3

(123−25, 123, 640)-Net over F₃ — Constructive and digital

Digital (98, 123, 640)-net over F₃, using

1 times m-reduction [i] based on digital (98, 124, 640)-net over F₃, using
- trace code for nets [i] based on digital (5, 31, 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]

(123−25, 123, 1579)-Net over F₃ — Digital

Digital (98, 123, 1579)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹²³, 1579, F₃, 25) (dual of [1579, 1456, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹²³, 2224, F₃, 25) (dual of [2224, 2101, 26]-code), using
  - constructionÂ X applied to C([0,12]) âŠ‚ C([0,9]) [i] based on
    1. 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]
    2. linear OA(3⁸⁵, 2188, F₃, 19) (dual of [2188, 2103, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188Â | 3¹⁴âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (BCH-bound) [i]
    3. linear OA(3¹⁰, 36, F₃, 5) (dual of [36, 26, 6]-code), using
      - (u,Â uâˆ’v,Â u+v+w)-construction [i] based on
        linear OA(3¹, 12, F₃, 1) (dual of [12, 11, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(3¹, s, F₃, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]
        linear OA(3³, 12, F₃, 2) (dual of [12, 9, 3]-code), using
        discarding factors / shortening the dual code based on linear OA(3³, 13, F₃, 2) (dual of [13, 10, 3]-code), using
        Hamming code H(3,3) [i]
        linear OA(3⁶, 12, F₃, 5) (dual of [12, 6, 6]-code), using
        extended Golay code G_e(3) [i]

(123−25, 123, 187515)-Net in Base 3 — Upper bound on s

There is no (98, 123, 187516)-net in base 3, because

1 times m-reduction [i] would yield (98, 122, 187516)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 16173 445248 701869 035627 359384 518786 764055 568168 682455 043393 > 3¹²² [i]

Best Known (123−25, 123, s)-Nets in Base 3

(123−25, 123, 640)-Net over F3 — Constructive and digital

(123−25, 123, 1579)-Net over F3 — Digital

(123−25, 123, 187515)-Net in Base 3 — Upper bound on s

(123−25, 123, 640)-Net over F₃ — Constructive and digital

(123−25, 123, 1579)-Net over F₃ — Digital