MinT - Best Known (123−21, 123, s)-Nets in Base 3

Best Known (123−21, 123, s)-Nets in Base 3

(123−21, 123, 695)-Net over F₃ — Constructive and digital

Digital (102, 123, 695)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (1, 11, 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 (91, 112, 688)-net over F₃, using
  - trace code for nets [i] based on digital (7, 28, 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]

(123−21, 123, 4572)-Net over F₃ — Digital

Digital (102, 123, 4572)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹²³, 4572, F₃, 21) (dual of [4572, 4449, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹²³, 6573, F₃, 21) (dual of [6573, 6450, 22]-code), using
  - constructionÂ X applied to C([0,10]) âŠ‚ C([1,10]) [i] based on
    1. linear OA(3¹¹³, 6562, F₃, 21) (dual of [6562, 6449, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562Â | 3¹⁶âˆ’1, defining interval IÂ =Â [0,10], and minimum distance dÂ â‰¥ |{−10,−9,…,10}|+1Â =Â 22 (BCH-bound) [i]
    2. linear OA(3¹¹², 6562, F₃, 10) (dual of [6562, 6450, 11]-code), using the narrow-sense BCH-code C(I) with length 6562Â | 3¹⁶âˆ’1, defining interval IÂ =Â [1,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]
    3. linear OA(3¹⁰, 11, F₃, 10) (dual of [11, 1, 11]-code or 11-arc in PG(9,3)), using
      - dual of repetition code with length 11 [i]

(123−21, 123, 1499073)-Net in Base 3 — Upper bound on s

There is no (102, 123, 1499074)-net in base 3, because

1 times m-reduction [i] would yield (102, 122, 1499074)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 16173 170462 957528 029086 510271 778720 093975 197404 124213 179621 > 3¹²² [i]