MinT - Best Known (123−22, 123, s)-Nets in Base 3

Best Known (123−22, 123, s)-Nets in Base 3

(123−22, 123, 688)-Net over F₃ — Constructive and digital

Digital (101, 123, 688)-net over F₃, using

t-expansion [i] based on digital (100, 123, 688)-net over F₃, using
- 1 times m-reduction [i] based on digital (100, 124, 688)-net over F₃, using
  - trace code for nets [i] based on digital (7, 31, 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]

(123−22, 123, 3360)-Net over F₃ — Digital

Digital (101, 123, 3360)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹²³, 3360, F₃, 22) (dual of [3360, 3237, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹²³, 6597, F₃, 22) (dual of [6597, 6474, 23]-code), using
  - constructionÂ X applied to C_e(21) âŠ‚ C_e(15) [i] based on
    1. linear OA(3¹¹³, 6561, F₃, 22) (dual of [6561, 6448, 23]-code), using an extension C_e(21) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 3⁸âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]
    2. linear OA(3⁸¹, 6561, F₃, 16) (dual of [6561, 6480, 17]-code), using an extension C_e(15) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 3⁸âˆ’1, defining interval IÂ =Â [1,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [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]

(123−22, 123, 530956)-Net in Base 3 — Upper bound on s

There is no (101, 123, 530957)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 48520 086249 291078 798194 069806 454884 178047 205019 558247 550955 > 3¹²³ [i]

Best Known (123−22, 123, s)-Nets in Base 3

(123−22, 123, 688)-Net over F3 — Constructive and digital

(123−22, 123, 3360)-Net over F3 — Digital

(123−22, 123, 530956)-Net in Base 3 — Upper bound on s

(123−22, 123, 688)-Net over F₃ — Constructive and digital

(123−22, 123, 3360)-Net over F₃ — Digital