MinT - Best Known (51, 51+48, s)-Nets in Base 32

Best Known (51, 51+48, s)-Nets in Base 32

(51, 51+48, 240)-Net over F₃₂ — Constructive and digital

Digital (51, 99, 240)-net over F₃₂, using

10 times m-reduction [i] based on digital (51, 109, 240)-net over F₃₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (11, 40, 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, 69, 120)-net over F₃₂, using
    - net from sequence [i] based on digital (11, 119)-sequence over F₃₂ (see above)

(51, 51+48, 513)-Net in Base 32 — Constructive

(51, 99, 513)-net in base 32, using

t-expansion [i] based on (46, 99, 513)-net in base 32, using
- 9 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]

(51, 51+48, 913)-Net over F₃₂ — Digital

Digital (51, 99, 913)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32⁹⁹, 913, F₃₂, 48) (dual of [913, 814, 49]-code), using
- discarding factors / shortening the dual code based on linear OA(32⁹⁹, 1047, F₃₂, 48) (dual of [1047, 948, 49]-code), using
  - constructionÂ X applied to C_e(47) âŠ‚ C_e(39) [i] based on
    1. linear OA(32⁹², 1024, F₃₂, 48) (dual of [1024, 932, 49]-code), using an extension C_e(47) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [1,47], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 48 [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(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]

(51, 51+48, 511368)-Net in Base 32 — Upper bound on s

There is no (51, 99, 511369)-net in base 32, because

the generalized Rao bound for nets shows that 32^mÂ â‰¥ 102294 122975 989332 869379 197393 455603 977939 202193 459209 381309 539945 277746 676215 402504 219507 868444 199065 639870 313809 619328 383241 873754 335358 003189 912664 > 32⁹⁹ [i]