MinT - Best Known (55, 55+52, s)-Nets in Base 32

Best Known (55, 55+52, s)-Nets in Base 32

(55, 55+52, 240)-Net over F₃₂ — Constructive and digital

Digital (55, 107, 240)-net over F₃₂, using

t-expansion [i] based on digital (51, 107, 240)-net over F₃₂, using
- 2 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)

(55, 55+52, 513)-Net in Base 32 — Constructive

(55, 107, 513)-net in base 32, using

t-expansion [i] based on (46, 107, 513)-net in base 32, using
- 1 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]

(55, 55+52, 952)-Net over F₃₂ — Digital

Digital (55, 107, 952)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32¹⁰⁷, 952, F₃₂, 52) (dual of [952, 845, 53]-code), using
- discarding factors / shortening the dual code based on linear OA(32¹⁰⁷, 1047, F₃₂, 52) (dual of [1047, 940, 53]-code), using
  - constructionÂ X applied to C_e(51) âŠ‚ C_e(43) [i] based on
    1. linear OA(32¹⁰⁰, 1024, F₃₂, 52) (dual of [1024, 924, 53]-code), using an extension C_e(51) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [1,51], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 52 [i]
    2. 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]
    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]

(55, 55+52, 532342)-Net in Base 32 — Upper bound on s

There is no (55, 107, 532343)-net in base 32, because

the generalized Rao bound for nets shows that 32^mÂ â‰¥ 112472 974075 459706 636040 226287 724079 063372 976961 920158 079684 478536 244111 065676 965963 052352 090986 875411 969366 898586 330279 632625 367029 791135 919984 582054 078748 221140 > 32¹⁰⁷ [i]