MinT - Best Known (49, 94, s)-Nets in Base 32

Best Known (49, 94, s)-Nets in Base 32

(49, 94, 240)-Net over F₃₂ — Constructive and digital

Digital (49, 94, 240)-net over F₃₂, using

9 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, 94, 513)-Net in Base 32 — Constructive

(49, 94, 513)-net in base 32, using

t-expansion [i] based on (46, 94, 513)-net in base 32, using
- 14 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, 94, 960)-Net over F₃₂ — Digital

Digital (49, 94, 960)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32⁹⁴, 960, F₃₂, 45) (dual of [960, 866, 46]-code), using
- discarding factors / shortening the dual code based on linear OA(32⁹⁴, 1042, F₃₂, 45) (dual of [1042, 948, 46]-code), using
  - constructionÂ X applied to C([0,22]) âŠ‚ C([0,19]) [i] based on
    1. linear OA(32⁸⁹, 1025, F₃₂, 45) (dual of [1025, 936, 46]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025Â | 32⁴âˆ’1, defining interval IÂ =Â [0,22], and minimum distance dÂ â‰¥ |{−22,−21,…,22}|+1Â =Â 46 (BCH-bound) [i]
    2. linear OA(32⁷⁷, 1025, F₃₂, 39) (dual of [1025, 948, 40]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025Â | 32⁴âˆ’1, defining interval IÂ =Â [0,19], and minimum distance dÂ â‰¥ |{−19,−18,…,19}|+1Â =Â 40 (BCH-bound) [i]
    3. linear OA(32⁵, 17, F₃₂, 5) (dual of [17, 12, 6]-code or 17-arc in PG(4,32)), using
      - discarding factors / shortening the dual code based on linear OA(32⁵, 32, F₃₂, 5) (dual of [32, 27, 6]-code or 32-arc in PG(4,32)), using
        Reedâ€“Solomon code RS(27,32) [i]

(49, 94, 673230)-Net in Base 32 — Upper bound on s

There is no (49, 94, 673231)-net in base 32, because

1 times m-reduction [i] would yield (49, 93, 673231)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 95 270858 686152 089132 743745 698342 184061 654154 300221 452953 319298 589914 713817 366608 693101 278318 759820 600074 622310 484628 881796 894519 923978 189012 > 32⁹³ [i]