MinT - Best Known (47, 47+46, s)-Nets in Base 32

Best Known (47, 47+46, s)-Nets in Base 32

(47, 47+46, 240)-Net over F₃₂ — Constructive and digital

Digital (47, 93, 240)-net over F₃₂, using

4 times m-reduction [i] based on digital (47, 97, 240)-net over F₃₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (11, 36, 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, 61, 120)-net over F₃₂, using
    - net from sequence [i] based on digital (11, 119)-sequence over F₃₂ (see above)

(47, 47+46, 513)-Net in Base 32 — Constructive

(47, 93, 513)-net in base 32, using

t-expansion [i] based on (46, 93, 513)-net in base 32, using
- 15 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]

(47, 47+46, 761)-Net over F₃₂ — Digital

Digital (47, 93, 761)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32⁹³, 761, F₃₂, 46) (dual of [761, 668, 47]-code), using
- discarding factors / shortening the dual code based on linear OA(32⁹³, 1041, F₃₂, 46) (dual of [1041, 948, 47]-code), using
  - constructionÂ X applied to C_e(45) âŠ‚ C_e(39) [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(32⁷⁶, 1024, F₃₂, 40) (dual of [1024, 948, 41]-code), using an extension C_e(39) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [1,39], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 40 [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]

(47, 47+46, 370784)-Net in Base 32 — Upper bound on s

There is no (47, 93, 370785)-net in base 32, because

the generalized Rao bound for nets shows that 32^mÂ â‰¥ 95 268745 363088 206191 107453 150916 909959 792111 217413 189271 721136 924898 223405 414350 893519 395010 180280 812808 735267 640706 051228 781657 934572 886688 > 32⁹³ [i]