MinT - Best Known (49, 93, s)-Nets in Base 32

Best Known (49, 93, s)-Nets in Base 32

(49, 93, 240)-Net over F₃₂ — Constructive and digital

Digital (49, 93, 240)-net over F₃₂, using

10 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, 93, 513)-Net in Base 32 — Constructive

(49, 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]

(49, 93, 1036)-Net over F₃₂ — Digital

Digital (49, 93, 1036)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32⁹³, 1036, F₃₂, 44) (dual of [1036, 943, 45]-code), using
- discarding factors / shortening the dual code based on linear OA(32⁹³, 1053, F₃₂, 44) (dual of [1053, 960, 45]-code), using
  - constructionÂ X applied to C_e(43) âŠ‚ C_e(33) [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(32⁶⁴, 1024, F₃₂, 34) (dual of [1024, 960, 35]-code), using an extension C_e(33) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [1,33], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 34 [i]
    3. linear OA(32⁹, 29, F₃₂, 9) (dual of [29, 20, 10]-code or 29-arc in PG(8,32)), using
      - discarding factors / shortening the dual code based on linear OA(32⁹, 32, F₃₂, 9) (dual of [32, 23, 10]-code or 32-arc in PG(8,32)), using
        Reedâ€“Solomon code RS(23,32) [i]

(49, 93, 673230)-Net in Base 32 — Upper bound on s

There is no (49, 93, 673231)-net in base 32, because

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]