MinT - Best Known (56−20, 56, s)-Nets in Base 32

Best Known (56−20, 56, s)-Nets in Base 32

(56−20, 56, 330)-Net over F₃₂ — Constructive and digital

Digital (36, 56, 330)-net over F₃₂, using

generalized (u,Â u+v)-construction [i] based on
1. digital (0, 2, 33)-net over F₃₂, using
  - kÂ =Â 2 construction [i]
2. digital (0, 2, 33)-net over F₃₂ (see above)
3. digital (0, 2, 33)-net over F₃₂ (see above)
4. digital (0, 2, 33)-net over F₃₂ (see above)
5. 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]
6. digital (0, 4, 33)-net over F₃₂, using
  - net from sequence [i] based on digital (0, 32)-sequence over F₃₂ (see above)
7. digital (0, 5, 33)-net over F₃₂, using
  - net from sequence [i] based on digital (0, 32)-sequence over F₃₂ (see above)
8. digital (0, 6, 33)-net over F₃₂, using
  - net from sequence [i] based on digital (0, 32)-sequence over F₃₂ (see above)
9. digital (0, 10, 33)-net over F₃₂, using
  - net from sequence [i] based on digital (0, 32)-sequence over F₃₂ (see above)
10. digital (0, 20, 33)-net over F₃₂, using
  - net from sequence [i] based on digital (0, 32)-sequence over F₃₂ (see above)

(56−20, 56, 1638)-Net in Base 32 — Constructive

(36, 56, 1638)-net in base 32, using

base change [i] based on digital (20, 40, 1638)-net over F₁₂₈, using
- 1 times m-reduction [i] based on digital (20, 41, 1638)-net over F₁₂₈, using
  - net defined by OOA [i] based on linear OOA(128⁴¹, 1638, F₁₂₈, 21, 21) (dual of [(1638, 21), 34357, 22]-NRT-code), using
    - OOA 10-folding and stacking with additional row [i] based on linear OA(128⁴¹, 16381, F₁₂₈, 21) (dual of [16381, 16340, 22]-code), using
      - discarding factors / shortening the dual code based on linear OA(128⁴¹, 16384, F₁₂₈, 21) (dual of [16384, 16343, 22]-code), using
        an extension C_e(20) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 128²âˆ’1, defining interval IÂ =Â [1,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]

(56−20, 56, 6993)-Net over F₃₂ — Digital

Digital (36, 56, 6993)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(56−20, 56, large)-Net in Base 32 — Upper bound on s

There is no (36, 56, large)-net in base 32, because

18 times m-reduction [i] would yield (36, 38, large)-net in base 32, but
- mutually orthogonal hypercube bound [i]