MinT - Best Known (74−37, 74, s)-Nets in Base 32

Best Known (74−37, 74, s)-Nets in Base 32

(74−37, 74, 218)-Net over F₃₂ — Constructive and digital

Digital (37, 74, 218)-net over F₃₂, using

1 times m-reduction [i] based on digital (37, 75, 218)-net over F₃₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (7, 26, 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]
  2. digital (11, 49, 120)-net over F₃₂, using
    - net from sequence [i] based on digital (11, 119)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 11 and N(F) ≥ 120, using
        a function field by SÃ©mirat [i]

(74−37, 74, 288)-Net in Base 32 — Constructive

(37, 74, 288)-net in base 32, using

24 times m-reduction [i] based on (37, 98, 288)-net in base 32, using
- base change [i] based on digital (9, 70, 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]

(74−37, 74, 603)-Net over F₃₂ — Digital

Digital (37, 74, 603)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32⁷⁴, 603, F₃₂, 37) (dual of [603, 529, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(32⁷⁴, 1030, F₃₂, 37) (dual of [1030, 956, 38]-code), using
  - constructionÂ X applied to C([0,18]) âŠ‚ C([0,17]) [i] based on
    1. linear OA(32⁷³, 1025, F₃₂, 37) (dual of [1025, 952, 38]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025Â | 32⁴âˆ’1, defining interval IÂ =Â [0,18], and minimum distance dÂ â‰¥ |{−18,−17,…,18}|+1Â =Â 38 (BCH-bound) [i]
    2. linear OA(32⁶⁹, 1025, F₃₂, 35) (dual of [1025, 956, 36]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025Â | 32⁴âˆ’1, defining interval IÂ =Â [0,17], and minimum distance dÂ â‰¥ |{−17,−16,…,17}|+1Â =Â 36 (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]

(74−37, 74, 309724)-Net in Base 32 — Upper bound on s

There is no (37, 74, 309725)-net in base 32, because

1 times m-reduction [i] would yield (37, 73, 309725)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 75 157333 573999 501783 522272 784704 353427 266593 456217 092123 750216 868321 557468 432810 539719 135576 177983 237931 114206 > 32⁷³ [i]