MinT - Best Known (109, 109+23, s)-Nets in Base 3

Best Known (109, 109+23, s)-Nets in Base 3

(109, 109+23, 695)-Net over F₃ — Constructive and digital

Digital (109, 132, 695)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (1, 12, 7)-net over F₃, using
  - net from sequence [i] based on digital (1, 6)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 1 and N(F) ≥ 7, using
      - an explicitly constructive algebraic function field [i]
2. digital (97, 120, 688)-net over F₃, using
  - trace code for nets [i] based on digital (7, 30, 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]

(109, 109+23, 4090)-Net over F₃ — Digital

Digital (109, 132, 4090)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹³², 4090, F₃, 23) (dual of [4090, 3958, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹³², 6573, F₃, 23) (dual of [6573, 6441, 24]-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OA(3¹¹, 12, F₃, 11) (dual of [12, 1, 12]-code or 12-arc in PG(10,3)), using
      - dual of repetition code with length 12 [i]
    2. linear OA(3¹²¹, 6561, F₃, 23) (dual of [6561, 6440, 24]-code), using
      - an extension C_e(22) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 3⁸âˆ’1, defining interval IÂ =Â [1,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]

(109, 109+23, 1180485)-Net in Base 3 — Upper bound on s

There is no (109, 132, 1180486)-net in base 3, because

1 times m-reduction [i] would yield (109, 131, 1180486)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 318 336161 961049 462508 503376 129034 963955 004080 425918 127077 036561 > 3¹³¹ [i]