MinT - Best Known (8, 19, s)-Nets in Base 32

Best Known (8, 19, s)-Nets in Base 32

(8, 19, 99)-Net over F₃₂ — Constructive and digital

Digital (8, 19, 99)-net over F₃₂, using

generalized (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 (0, 5, 33)-net over F₃₂, using
  - net from sequence [i] based on digital (0, 32)-sequence over F₃₂ (see above)
3. digital (0, 11, 33)-net over F₃₂, using
  - net from sequence [i] based on digital (0, 32)-sequence over F₃₂ (see above)

(8, 19, 113)-Net over F₃₂ — Digital

Digital (8, 19, 113)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32¹⁹, 113, F₃₂, 11) (dual of [113, 94, 12]-code), using
- 27 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 5 times 0, 1, 20 times 0) [i] based on linear OA(32¹⁶, 83, F₃₂, 11) (dual of [83, 67, 12]-code), using
  - extended algebraic-geometric code AG_e(F,71P) [i] based on function field F/F₃₂ with g(F) = 5 and N(F) ≥ 83, using
    - a curve from the manYPoints database [i]

(8, 19, 257)-Net in Base 32 — Constructive

(8, 19, 257)-net in base 32, using

2 times m-reduction [i] based on (8, 21, 257)-net in base 32, using
- base change [i] based on (2, 15, 257)-net in base 128, using
  - 1 times m-reduction [i] based on (2, 16, 257)-net in base 128, using
    - base change [i] based on digital (0, 14, 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]

(8, 19, 22027)-Net in Base 32 — Upper bound on s

There is no (8, 19, 22028)-net in base 32, because

1 times m-reduction [i] would yield (8, 18, 22028)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 1237 985622 731679 348927 463602 > 32¹⁸ [i]