MinT - Best Known (35−13, 35, s)-Nets in Base 32

Best Known (35−13, 35, s)-Nets in Base 32

(35−13, 35, 363)-Net over F₃₂ — Constructive and digital

Digital (22, 35, 363)-net over F₃₂, using

generalized (u,Â u+v)-construction [i] based on
1. digital (0, 1, 33)-net over F₃₂, using
  - s-reduction based on digital (0, 1, s)-net over F₃₂ with arbitrarily large s, using
    - construction for strength kÂ =Â 1 [i]
2. digital (0, 1, 33)-net over F₃₂ (see above)
3. digital (0, 1, 33)-net over F₃₂ (see above)
4. digital (0, 1, 33)-net over F₃₂ (see above)
5. digital (0, 1, 33)-net over F₃₂ (see above)
6. digital (0, 2, 33)-net over F₃₂, using
  - kÂ =Â 2 construction [i]
7. digital (0, 2, 33)-net over F₃₂ (see above)
8. 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]
9. digital (0, 4, 33)-net over F₃₂, using
  - net from sequence [i] based on digital (0, 32)-sequence over F₃₂ (see above)
10. digital (0, 6, 33)-net over F₃₂, using
  - net from sequence [i] based on digital (0, 32)-sequence over F₃₂ (see above)
11. digital (0, 13, 33)-net over F₃₂, using
  - net from sequence [i] based on digital (0, 32)-sequence over F₃₂ (see above)

(35−13, 35, 2730)-Net in Base 32 — Constructive

(22, 35, 2730)-net in base 32, using

base change [i] based on digital (12, 25, 2730)-net over F₁₂₈, using
- net defined by OOA [i] based on linear OOA(128²⁵, 2730, F₁₂₈, 13, 13) (dual of [(2730, 13), 35465, 14]-NRT-code), using
  - OOA 6-folding and stacking with additional row [i] based on linear OA(128²⁵, 16381, F₁₂₈, 13) (dual of [16381, 16356, 14]-code), using
    - discarding factors / shortening the dual code based on linear OA(128²⁵, 16384, F₁₂₈, 13) (dual of [16384, 16359, 14]-code), using
      - an extension C_e(12) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 128²âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

(35−13, 35, 4194)-Net over F₃₂ — Digital

Digital (22, 35, 4194)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(35−13, 35, 4619)-Net in Base 32

(22, 35, 4619)-net in base 32, using

base change [i] based on digital (12, 25, 4619)-net over F₁₂₈, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(128²⁵, 4619, F₁₂₈, 3, 13) (dual of [(4619, 3), 13832, 14]-NRT-code), using
  - discarding factors / shortening the dual code based on linear OOA(128²⁵, 5462, F₁₂₈, 3, 13) (dual of [(5462, 3), 16361, 14]-NRT-code), using
    - OOA 3-folding [i] based on linear OA(128²⁵, 16386, F₁₂₈, 13) (dual of [16386, 16361, 14]-code), using
      - constructionÂ X applied to C_e(12) âŠ‚ C_e(11) [i] based on
        linear OA(128²⁵, 16384, F₁₂₈, 13) (dual of [16384, 16359, 14]-code), using an extension C_e(12) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 128²âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]
        linear OA(128²³, 16384, F₁₂₈, 12) (dual of [16384, 16361, 13]-code), using an extension C_e(11) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 128²âˆ’1, defining interval IÂ =Â [1,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]
        linear OA(128⁰, 2, F₁₂₈, 0) (dual of [2, 2, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(128⁰, s, F₁₂₈, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(35−13, 35, large)-Net in Base 32 — Upper bound on s

There is no (22, 35, large)-net in base 32, because

11 times m-reduction [i] would yield (22, 24, large)-net in base 32, but
- mutually orthogonal hypercube bound [i]