MinT - Best Known (49, 49+47, s)-Nets in Base 32

Best Known (49, 49+47, s)-Nets in Base 32

(49, 49+47, 240)-Net over F₃₂ — Constructive and digital

Digital (49, 96, 240)-net over F₃₂, using

7 times m-reduction [i] based on digital (49, 103, 240)-net over F₃₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (11, 38, 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]
  2. digital (11, 65, 120)-net over F₃₂, using
    - net from sequence [i] based on digital (11, 119)-sequence over F₃₂ (see above)

(49, 49+47, 513)-Net in Base 32 — Constructive

(49, 96, 513)-net in base 32, using

t-expansion [i] based on (46, 96, 513)-net in base 32, using
- 12 times m-reduction [i] based on (46, 108, 513)-net in base 32, using
  - base change [i] based on digital (28, 90, 513)-net over F₆₄, using
    - net from sequence [i] based on digital (28, 512)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 28 and N(F) ≥ 513, using
        the Hermitian function field over F₆₄ [i]

(49, 49+47, 835)-Net over F₃₂ — Digital

Digital (49, 96, 835)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32⁹⁶, 835, F₃₂, 47) (dual of [835, 739, 48]-code), using
- discarding factors / shortening the dual code based on linear OA(32⁹⁶, 1036, F₃₂, 47) (dual of [1036, 940, 48]-code), using
  - constructionÂ X applied to C([0,23]) âŠ‚ C([0,21]) [i] based on
    1. linear OA(32⁹³, 1025, F₃₂, 47) (dual of [1025, 932, 48]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025Â | 32⁴âˆ’1, defining interval IÂ =Â [0,23], and minimum distance dÂ â‰¥ |{−23,−22,…,23}|+1Â =Â 48 (BCH-bound) [i]
    2. linear OA(32⁸⁵, 1025, F₃₂, 43) (dual of [1025, 940, 44]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025Â | 32⁴âˆ’1, defining interval IÂ =Â [0,21], and minimum distance dÂ â‰¥ |{−21,−20,…,21}|+1Â =Â 44 (BCH-bound) [i]
    3. linear OA(32³, 11, F₃₂, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,32) or 11-cap in PG(2,32)), using
      - discarding factors / shortening the dual code based on linear OA(32³, 32, F₃₂, 3) (dual of [32, 29, 4]-code or 32-arc in PG(2,32) or 32-cap in PG(2,32)), using
        Reedâ€“Solomon code RS(29,32) [i]

(49, 49+47, 501196)-Net in Base 32 — Upper bound on s

There is no (49, 96, 501197)-net in base 32, because

1 times m-reduction [i] would yield (49, 95, 501197)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 97555 820221 894056 886930 066955 076629 724870 477373 207223 925166 222453 040696 317985 204016 081857 134176 388271 957536 911562 909696 876648 206816 721389 175200 > 32⁹⁵ [i]