MinT - Best Known (50, 97, s)-Nets in Base 32

Best Known (50, 97, s)-Nets in Base 32

(50, 97, 240)-Net over F₃₂ — Constructive and digital

Digital (50, 97, 240)-net over F₃₂, using

9 times m-reduction [i] based on digital (50, 106, 240)-net over F₃₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (11, 39, 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, 67, 120)-net over F₃₂, using
    - net from sequence [i] based on digital (11, 119)-sequence over F₃₂ (see above)

(50, 97, 513)-Net in Base 32 — Constructive

(50, 97, 513)-net in base 32, using

t-expansion [i] based on (46, 97, 513)-net in base 32, using
- 11 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]

(50, 97, 903)-Net over F₃₂ — Digital

Digital (50, 97, 903)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32⁹⁷, 903, F₃₂, 47) (dual of [903, 806, 48]-code), using
- discarding factors / shortening the dual code based on linear OA(32⁹⁷, 1047, F₃₂, 47) (dual of [1047, 950, 48]-code), using
  - constructionÂ X applied to C_e(46) âŠ‚ C_e(38) [i] based on
    1. linear OA(32⁹⁰, 1024, F₃₂, 47) (dual of [1024, 934, 48]-code), using an extension C_e(46) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [1,46], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 47 [i]
    2. linear OA(32⁷⁴, 1024, F₃₂, 39) (dual of [1024, 950, 40]-code), using an extension C_e(38) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [1,38], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 39 [i]
    3. linear OA(32⁷, 23, F₃₂, 7) (dual of [23, 16, 8]-code or 23-arc in PG(6,32)), using
      - discarding factors / shortening the dual code based on linear OA(32⁷, 32, F₃₂, 7) (dual of [32, 25, 8]-code or 32-arc in PG(6,32)), using
        Reedâ€“Solomon code RS(25,32) [i]

(50, 97, 582707)-Net in Base 32 — Upper bound on s

There is no (50, 97, 582708)-net in base 32, because

1 times m-reduction [i] would yield (50, 96, 582708)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 3 121751 721152 771101 016801 888303 450110 420037 726664 243293 959760 256862 397791 834340 175620 726983 153854 030406 788352 145486 401427 071266 345925 171057 843784 > 32⁹⁶ [i]