MinT - Best Known (222−45, 222, s)-Nets in Base 3

Best Known (222−45, 222, s)-Nets in Base 3

(222−45, 222, 688)-Net over F₃ — Constructive and digital

Digital (177, 222, 688)-net over F₃, using

t-expansion [i] based on digital (175, 222, 688)-net over F₃, using
- 2 times m-reduction [i] based on digital (175, 224, 688)-net over F₃, using
  - trace code for nets [i] based on digital (7, 56, 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]

(222−45, 222, 2289)-Net over F₃ — Digital

Digital (177, 222, 2289)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3²²², 2289, F₃, 45) (dual of [2289, 2067, 46]-code), using
- 76 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (3, 1, 0, 1, 0, 0, 1, 4 times 0, 1, 8 times 0, 1, 12 times 0, 1, 17 times 0, 1, 24 times 0) [i] based on linear OA(3²¹², 2203, F₃, 45) (dual of [2203, 1991, 46]-code), using
  - constructionÂ X applied to C([0,22]) âŠ‚ C([0,21]) [i] based on
    1. 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]
    2. linear OA(3¹⁹⁷, 2188, F₃, 43) (dual of [2188, 1991, 44]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188Â | 3¹⁴âˆ’1, defining interval IÂ =Â [0,21], and minimum distance dÂ â‰¥ |{−21,−20,…,21}|+1Â =Â 44 (BCH-bound) [i]
    3. linear OA(3¹, 15, F₃, 1) (dual of [15, 14, 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]

(222−45, 222, 280989)-Net in Base 3 — Upper bound on s

There is no (177, 222, 280990)-net in base 3, because

1 times m-reduction [i] would yield (177, 221, 280990)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2778 582769 685705 302914 499868 822989 994502 021340 823434 054893 455680 410522 577910 372593 487946 526718 104057 778949 > 3²²¹ [i]