MinT - Best Known (54, 54+52, s)-Nets in Base 32

Best Known (54, 54+52, s)-Nets in Base 32

(54, 54+52, 240)-Net over F₃₂ — Constructive and digital

Digital (54, 106, 240)-net over F₃₂, using

t-expansion [i] based on digital (51, 106, 240)-net over F₃₂, using
- 3 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)

(54, 54+52, 513)-Net in Base 32 — Constructive

(54, 106, 513)-net in base 32, using

t-expansion [i] based on (46, 106, 513)-net in base 32, using
- 2 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]

(54, 54+52, 887)-Net over F₃₂ — Digital

Digital (54, 106, 887)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32¹⁰⁶, 887, F₃₂, 52) (dual of [887, 781, 53]-code), using
- discarding factors / shortening the dual code based on linear OA(32¹⁰⁶, 1044, F₃₂, 52) (dual of [1044, 938, 53]-code), using
  - constructionÂ X applied to C_e(51) âŠ‚ C_e(44) [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(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]
    3. linear OA(32⁶, 20, F₃₂, 6) (dual of [20, 14, 7]-code or 20-arc in PG(5,32)), using
      - discarding factors / shortening the dual code based on linear OA(32⁶, 32, F₃₂, 6) (dual of [32, 26, 7]-code or 32-arc in PG(5,32)), using
        Reedâ€“Solomon code RS(26,32) [i]

(54, 54+52, 465907)-Net in Base 32 — Upper bound on s

There is no (54, 106, 465908)-net in base 32, because

the generalized Rao bound for nets shows that 32^mÂ â‰¥ 3514 897302 393823 614747 079135 004435 860986 870068 984370 520402 219024 876561 764169 269595 660723 981756 447300 012245 815123 792439 382184 550646 792832 622020 937772 959453 924172 > 32¹⁰⁶ [i]