MinT - Best Known (51, 51+49, s)-Nets in Base 32

Best Known (51, 51+49, s)-Nets in Base 32

(51, 51+49, 240)-Net over F₃₂ — Constructive and digital

Digital (51, 100, 240)-net over F₃₂, using

9 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+49, 513)-Net in Base 32 — Constructive

(51, 100, 513)-net in base 32, using

t-expansion [i] based on (46, 100, 513)-net in base 32, using
- 8 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+49, 856)-Net over F₃₂ — Digital

Digital (51, 100, 856)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32¹⁰⁰, 856, F₃₂, 49) (dual of [856, 756, 50]-code), using
- discarding factors / shortening the dual code based on linear OA(32¹⁰⁰, 1036, F₃₂, 49) (dual of [1036, 936, 50]-code), using
  - constructionÂ X applied to C([0,24]) âŠ‚ C([0,22]) [i] based on
    1. linear OA(32⁹⁷, 1025, F₃₂, 49) (dual of [1025, 928, 50]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025Â | 32⁴âˆ’1, defining interval IÂ =Â [0,24], and minimum distance dÂ â‰¥ |{−24,−23,…,24}|+1Â =Â 50 (BCH-bound) [i]
    2. linear OA(32⁸⁹, 1025, F₃₂, 45) (dual of [1025, 936, 46]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025Â | 32⁴âˆ’1, defining interval IÂ =Â [0,22], and minimum distance dÂ â‰¥ |{−22,−21,…,22}|+1Â =Â 46 (BCH-bound) [i]
    3. linear OA(32³, 11, F₃₂, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,32) or 11-cap in PG(2,32)), using
      - discarding factors / shortening the dual code based on linear OA(32³, 32, F₃₂, 3) (dual of [32, 29, 4]-code or 32-arc in PG(2,32) or 32-cap in PG(2,32)), using
        Reedâ€“Solomon code RS(29,32) [i]

(51, 51+49, 511368)-Net in Base 32 — Upper bound on s

There is no (51, 100, 511369)-net in base 32, because

1 times m-reduction [i] would yield (51, 99, 511369)-net in base 32, but
- 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]