MinT - Best Known (10−6, 10, s)-Nets in Base 32

Best Known (10−6, 10, s)-Nets in Base 32

(10−6, 10, 77)-Net over F₃₂ — Constructive and digital

Digital (4, 10, 77)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (0, 3, 33)-net over F₃₂, using
  - net from sequence [i] based on digital (0, 32)-sequence over F₃₂, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 0 and N(F) ≥ 33, using
      - the rational function field F₃₂(x) [i]
    - Niederreiter sequence [i]
2. digital (1, 7, 44)-net over F₃₂, using
  - net from sequence [i] based on digital (1, 43)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 1 and N(F) ≥ 44, using
      - a function field by SÃ©mirat [i]

(10−6, 10, 97)-Net over F₃₂ — Digital

Digital (4, 10, 97)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32¹⁰, 97, F₃₂, 6) (dual of [97, 87, 7]-code), using
- constructionÂ XX applied to C₁Â =Â C([92,3]), C₂Â =Â C([0,4]), C₃Â =Â C₁Â +Â C₂Â =Â C([0,3]), and C_âˆ©Â =Â C₁Â âˆ©Â C₂Â =Â C([92,4]) [i] based on
  1. linear OA(32⁸, 93, F₃₂, 5) (dual of [93, 85, 6]-code), using the BCH-code C(I) with length 93Â | 32²âˆ’1, defining interval IÂ =Â {−1,0,1,2,3}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
  2. linear OA(32⁸, 93, F₃₂, 5) (dual of [93, 85, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 93Â | 32²âˆ’1, defining interval IÂ =Â [0,4], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
  3. linear OA(32¹⁰, 93, F₃₂, 6) (dual of [93, 83, 7]-code), using the BCH-code C(I) with length 93Â | 32²âˆ’1, defining interval IÂ =Â {−1,0,…,4}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
  4. linear OA(32⁶, 93, F₃₂, 4) (dual of [93, 87, 5]-code), using the expurgated narrow-sense BCH-code C(I) with length 93Â | 32²âˆ’1, defining interval IÂ =Â [0,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]
  5. linear OA(32⁰, 2, F₃₂, 0) (dual of [2, 2, 1]-code), using
    - discarding factors / shortening the dual code based on linear OA(32⁰, s, F₃₂, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
      - OA with only one run / dual of code without redundancy [i]
  6. linear OA(32⁰, 2, F₃₂, 0) (dual of [2, 2, 1]-code) (see above)

(10−6, 10, 129)-Net in Base 32 — Constructive

(4, 10, 129)-net in base 32, using

4 times m-reduction [i] based on (4, 14, 129)-net in base 32, using
- base change [i] based on digital (0, 10, 129)-net over F₁₂₈, using
  - net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 0 and N(F) ≥ 129, using
      - the rational function field F₁₂₈(x) [i]
    - Niederreiter sequence [i]

(10−6, 10, 6096)-Net in Base 32 — Upper bound on s

There is no (4, 10, 6097)-net in base 32, because

the generalized Rao bound for nets shows that 32^mÂ â‰¥ 1125 943900 014480 > 32¹⁰ [i]

Best Known (10−6, 10, s)-Nets in Base 32

(10−6, 10, 77)-Net over F32 — Constructive and digital

(10−6, 10, 97)-Net over F32 — Digital

(10−6, 10, 129)-Net in Base 32 — Constructive

(10−6, 10, 6096)-Net in Base 32 — Upper bound on s

(10−6, 10, 77)-Net over F₃₂ — Constructive and digital

(10−6, 10, 97)-Net over F₃₂ — Digital