MinT - Best Known (96−46, 96, s)-Nets in Base 32

Best Known (96−46, 96, s)-Nets in Base 32

(96−46, 96, 240)-Net over F₃₂ — Constructive and digital

Digital (50, 96, 240)-net over F₃₂, using

10 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)

(96−46, 96, 513)-Net in Base 32 — Constructive

(50, 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]

(96−46, 96, 969)-Net over F₃₂ — Digital

Digital (50, 96, 969)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32⁹⁶, 969, F₃₂, 46) (dual of [969, 873, 47]-code), using
- discarding factors / shortening the dual code based on linear OA(32⁹⁶, 1050, F₃₂, 46) (dual of [1050, 954, 47]-code), using
  - constructionÂ X applied to C_e(45) âŠ‚ C_e(36) [i] based on
    1. linear OA(32⁸⁸, 1024, F₃₂, 46) (dual of [1024, 936, 47]-code), using an extension C_e(45) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [1,45], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 46 [i]
    2. linear OA(32⁷⁰, 1024, F₃₂, 37) (dual of [1024, 954, 38]-code), using an extension C_e(36) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [1,36], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 37 [i]
    3. linear OA(32⁸, 26, F₃₂, 8) (dual of [26, 18, 9]-code or 26-arc in PG(7,32)), using
      - discarding factors / shortening the dual code based on linear OA(32⁸, 32, F₃₂, 8) (dual of [32, 24, 9]-code or 32-arc in PG(7,32)), using
        Reedâ€“Solomon code RS(24,32) [i]

(96−46, 96, 582707)-Net in Base 32 — Upper bound on s

There is no (50, 96, 582708)-net in base 32, because

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]