MinT - Best Known (48, 48+45, s)-Nets in Base 32

Best Known (48, 48+45, s)-Nets in Base 32

(48, 48+45, 240)-Net over F₃₂ — Constructive and digital

Digital (48, 93, 240)-net over F₃₂, using

7 times m-reduction [i] based on digital (48, 100, 240)-net over F₃₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (11, 37, 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, 63, 120)-net over F₃₂, using
    - net from sequence [i] based on digital (11, 119)-sequence over F₃₂ (see above)

(48, 48+45, 513)-Net in Base 32 — Constructive

(48, 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]

(48, 48+45, 885)-Net over F₃₂ — Digital

Digital (48, 93, 885)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32⁹³, 885, F₃₂, 45) (dual of [885, 792, 46]-code), using
- discarding factors / shortening the dual code based on linear OA(32⁹³, 1047, F₃₂, 45) (dual of [1047, 954, 46]-code), using
  - constructionÂ X applied to C_e(44) âŠ‚ C_e(36) [i] based on
    1. linear OA(32⁸⁶, 1024, F₃₂, 45) (dual of [1024, 938, 46]-code), using an extension C_e(44) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [1,44], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 45 [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(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]

(48, 48+45, 575104)-Net in Base 32 — Upper bound on s

There is no (48, 93, 575105)-net in base 32, because

1 times m-reduction [i] would yield (48, 92, 575105)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 2 977241 441482 558094 820392 482258 212708 588178 765857 202265 509356 577570 494147 920921 849896 934035 299690 321935 382162 275651 143927 912879 831854 098496 > 32⁹² [i]