MinT - Best Known (47, 47+44, s)-Nets in Base 32

Best Known (47, 47+44, s)-Nets in Base 32

(47, 47+44, 240)-Net over F₃₂ — Constructive and digital

Digital (47, 91, 240)-net over F₃₂, using

6 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+44, 513)-Net in Base 32 — Constructive

(47, 91, 513)-net in base 32, using

t-expansion [i] based on (46, 91, 513)-net in base 32, using
- 17 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+44, 875)-Net over F₃₂ — Digital

Digital (47, 91, 875)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32⁹¹, 875, F₃₂, 44) (dual of [875, 784, 45]-code), using
- discarding factors / shortening the dual code based on linear OA(32⁹¹, 1047, F₃₂, 44) (dual of [1047, 956, 45]-code), using
  - constructionÂ X applied to C_e(43) âŠ‚ C_e(35) [i] based on
    1. linear OA(32⁸⁴, 1024, F₃₂, 44) (dual of [1024, 940, 45]-code), using an extension C_e(43) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [1,43], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 44 [i]
    2. linear OA(32⁶⁸, 1024, F₃₂, 36) (dual of [1024, 956, 37]-code), using an extension C_e(35) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [1,35], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 36 [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]

(47, 47+44, 491279)-Net in Base 32 — Upper bound on s

There is no (47, 91, 491280)-net in base 32, because

the generalized Rao bound for nets shows that 32^mÂ â‰¥ 93035 464837 121072 402199 785711 696823 680364 492259 056724 017359 126434 453515 041579 347249 967783 820031 968027 516530 234091 474552 777133 185460 973616 > 32⁹¹ [i]