MinT - Best Known (23, 23+24, s)-Nets in Base 32

Best Known (23, 23+24, s)-Nets in Base 32

(23, 23+24, 162)-Net over F₃₂ — Constructive and digital

Digital (23, 47, 162)-net over F₃₂, using

2 times m-reduction [i] based on digital (23, 49, 162)-net over F₃₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (3, 16, 64)-net over F₃₂, using
    - net from sequence [i] based on digital (3, 63)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 3 and N(F) ≥ 64, using
        a function field by SÃ©mirat [i]
  2. digital (7, 33, 98)-net over F₃₂, using
    - net from sequence [i] based on digital (7, 97)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 7 and N(F) ≥ 98, using
        a function field by SÃ©mirat [i]

(23, 23+24, 288)-Net in Base 32 — Constructive

(23, 47, 288)-net in base 32, using

2 times m-reduction [i] based on (23, 49, 288)-net in base 32, using
- base change [i] based on digital (9, 35, 288)-net over F₁₂₈, using
  - net from sequence [i] based on digital (9, 287)-sequence over F₁₂₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 9 and N(F) ≥ 288, using
      - a function field by SÃ©mirat [i]

(23, 23+24, 460)-Net over F₃₂ — Digital

Digital (23, 47, 460)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(32⁴⁷, 460, F₃₂, 2, 24) (dual of [(460, 2), 873, 25]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(32⁴⁷, 513, F₃₂, 2, 24) (dual of [(513, 2), 979, 25]-NRT-code), using
  - OOA 2-folding [i] based on linear OA(32⁴⁷, 1026, F₃₂, 24) (dual of [1026, 979, 25]-code), using
    - constructionÂ X applied to C_e(23) âŠ‚ C_e(22) [i] based on
      1. linear OA(32⁴⁷, 1024, F₃₂, 24) (dual of [1024, 977, 25]-code), using an extension C_e(23) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [1,23], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 24 [i]
      2. linear OA(32⁴⁵, 1024, F₃₂, 23) (dual of [1024, 979, 24]-code), using an extension C_e(22) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [1,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]
      3. 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]

(23, 23+24, 134014)-Net in Base 32 — Upper bound on s

There is no (23, 47, 134015)-net in base 32, because

the generalized Rao bound for nets shows that 32^mÂ â‰¥ 55217 052287 631660 842194 488532 088794 091007 903035 728432 812905 906719 433373 > 32⁴⁷ [i]