MinT - Best Known (25−11, 25, s)-Nets in Base 32

Best Known (25−11, 25, s)-Nets in Base 32

(25−11, 25, 231)-Net over F₃₂ — Constructive and digital

Digital (14, 25, 231)-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, 2, 33)-net over F₃₂, using
  - kÂ =Â 2 construction [i]
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, 5, 33)-net over F₃₂, using
  - net from sequence [i] based on digital (0, 32)-sequence over F₃₂ (see above)
7. digital (0, 11, 33)-net over F₃₂, using
  - net from sequence [i] based on digital (0, 32)-sequence over F₃₂ (see above)

(25−11, 25, 514)-Net in Base 32 — Constructive

(14, 25, 514)-net in base 32, using

(u,Â u+v)-construction [i] based on
1. digital (2, 7, 496)-net over F₃₂, using
  - net defined by OOA [i] based on linear OOA(32⁷, 496, F₃₂, 5, 5) (dual of [(496, 5), 2473, 6]-NRT-code), using
    - OOA 2-folding and stacking with additional row [i] based on linear OA(32⁷, 993, F₃₂, 5) (dual of [993, 986, 6]-code), using
      - a code by Danev and Olsson [i]
2. (7, 18, 257)-net in base 32, using
  - base change [i] based on (4, 15, 257)-net in base 64, using
    - 1 times m-reduction [i] based on (4, 16, 257)-net in base 64, using
      - base change [i] based on digital (0, 12, 257)-net over F₂₅₆, using
        net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
        Niederreiter sequence [i]

(25−11, 25, 1084)-Net over F₃₂ — Digital

Digital (14, 25, 1084)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32²⁵, 1084, F₃₂, 11) (dual of [1084, 1059, 12]-code), using
- 51 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 7 times 0, 1, 42 times 0) [i] based on linear OA(32²², 1030, F₃₂, 11) (dual of [1030, 1008, 12]-code), using
  - constructionÂ X applied to C([0,5]) âŠ‚ C([0,4]) [i] based on
    1. linear OA(32²¹, 1025, F₃₂, 11) (dual of [1025, 1004, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025Â | 32⁴âˆ’1, defining interval IÂ =Â [0,5], and minimum distance dÂ â‰¥ |{−5,−4,…,5}|+1Â =Â 12 (BCH-bound) [i]
    2. linear OA(32¹⁷, 1025, F₃₂, 9) (dual of [1025, 1008, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025Â | 32⁴âˆ’1, defining interval IÂ =Â [0,4], and minimum distance dÂ â‰¥ |{−4,−3,…,4}|+1Â =Â 10 (BCH-bound) [i]
    3. linear OA(32¹, 5, F₃₂, 1) (dual of [5, 4, 2]-code), using
      - discarding factors / shortening the dual code based on linear OA(32¹, s, F₃₂, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(25−11, 25, 1409917)-Net in Base 32 — Upper bound on s

There is no (14, 25, 1409918)-net in base 32, because

1 times m-reduction [i] would yield (14, 24, 1409918)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 1 329229 059448 361263 411661 788214 048565 > 32²⁴ [i]