MinT - Best Known (50−25, 50, s)-Nets in Base 32

Best Known (50−25, 50, s)-Nets in Base 32

(50−25, 50, 174)-Net over F₃₂ — Constructive and digital

Digital (25, 50, 174)-net over F₃₂, using

1 times m-reduction [i] based on digital (25, 51, 174)-net over F₃₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (5, 18, 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, 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]

(50−25, 50, 288)-Net in Base 32 — Constructive

(25, 50, 288)-net in base 32, using

6 times m-reduction [i] based on (25, 56, 288)-net in base 32, using
- base change [i] based on digital (9, 40, 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]

(50−25, 50, 515)-Net over F₃₂ — Digital

Digital (25, 50, 515)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(32⁵⁰, 515, F₃₂, 2, 25) (dual of [(515, 2), 980, 26]-NRT-code), using
- OOA 2-folding [i] based on linear OA(32⁵⁰, 1030, F₃₂, 25) (dual of [1030, 980, 26]-code), using
  - constructionÂ X applied to C([0,12]) âŠ‚ C([0,11]) [i] based on
    1. linear OA(32⁴⁹, 1025, F₃₂, 25) (dual of [1025, 976, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025Â | 32⁴âˆ’1, defining interval IÂ =Â [0,12], and minimum distance dÂ â‰¥ |{−12,−11,…,12}|+1Â =Â 26 (BCH-bound) [i]
    2. 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]
    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]

(50−25, 50, 238791)-Net in Base 32 — Upper bound on s

There is no (25, 50, 238792)-net in base 32, because

1 times m-reduction [i] would yield (25, 49, 238792)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 56 541733 435692 859781 145113 951073 086822 385027 532869 223605 297809 936206 496232 > 32⁴⁹ [i]