MinT - Best Known (46−23, 46, s)-Nets in Base 32

Best Known (46−23, 46, s)-Nets in Base 32

(46−23, 46, 174)-Net over F₃₂ — Constructive and digital

Digital (23, 46, 174)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (5, 16, 76)-net over F₃₂, using
  - net from sequence [i] based on digital (5, 75)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 5 and N(F) ≥ 76, using
      - a function field by SÃ©mirat [i]
2. digital (7, 30, 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]

(46−23, 46, 288)-Net in Base 32 — Constructive

(23, 46, 288)-net in base 32, using

3 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]

(46−23, 46, 515)-Net over F₃₂ — Digital

Digital (23, 46, 515)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(32⁴⁶, 515, F₃₂, 2, 23) (dual of [(515, 2), 984, 24]-NRT-code), using
- OOA 2-folding [i] based on linear OA(32⁴⁶, 1030, F₃₂, 23) (dual of [1030, 984, 24]-code), using
  - constructionÂ X applied to C([0,11]) âŠ‚ C([0,10]) [i] based on
    1. linear OA(32⁴⁵, 1025, F₃₂, 23) (dual of [1025, 980, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025Â | 32⁴âˆ’1, defining interval IÂ =Â [0,11], and minimum distance dÂ â‰¥ |{−11,−10,…,11}|+1Â =Â 24 (BCH-bound) [i]
    2. linear OA(32⁴¹, 1025, F₃₂, 21) (dual of [1025, 984, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025Â | 32⁴âˆ’1, defining interval IÂ =Â [0,10], and minimum distance dÂ â‰¥ |{−10,−9,…,10}|+1Â =Â 22 (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]

(46−23, 46, 227548)-Net in Base 32 — Upper bound on s

There is no (23, 46, 227549)-net in base 32, because

1 times m-reduction [i] would yield (23, 45, 227549)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 53 921203 563966 599857 107123 316400 528802 612863 583006 826719 435284 156060 > 32⁴⁵ [i]