MinT - Best Known (19, 31, s)-Nets in Base 32

Best Known (19, 31, s)-Nets in Base 32

(19, 31, 264)-Net over F₃₂ — Constructive and digital

Digital (19, 31, 264)-net over F₃₂, using

1 times m-reduction [i] based on digital (19, 32, 264)-net over F₃₂, using
- generalized (u,Â u+v)-construction [i] based on
  1. digital (0, 1, 33)-net over F₃₂, using
    - s-reduction based on digital (0, 1, s)-net over F₃₂ with arbitrarily large s, using
      - construction for strength kÂ =Â 1 [i]
  2. digital (0, 1, 33)-net over F₃₂ (see above)
  3. digital (0, 2, 33)-net over F₃₂, using
    - kÂ =Â 2 construction [i]
  4. digital (0, 2, 33)-net over F₃₂ (see above)
  5. digital (0, 3, 33)-net over F₃₂, using
    - net from sequence [i] based on digital (0, 32)-sequence over F₃₂, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 0 and N(F) ≥ 33, using
        the rational function field F₃₂(x) [i]
      - Niederreiter sequence [i]
  6. digital (0, 4, 33)-net over F₃₂, using
    - net from sequence [i] based on digital (0, 32)-sequence over F₃₂ (see above)
  7. digital (0, 6, 33)-net over F₃₂, using
    - net from sequence [i] based on digital (0, 32)-sequence over F₃₂ (see above)
  8. digital (0, 13, 33)-net over F₃₂, using
    - net from sequence [i] based on digital (0, 32)-sequence over F₃₂ (see above)

(19, 31, 684)-Net in Base 32 — Constructive

(19, 31, 684)-net in base 32, using

32¹ times duplication [i] based on (18, 30, 684)-net in base 32, using
- base change [i] based on digital (13, 25, 684)-net over F₆₄, using
  - net defined by OOA [i] based on linear OOA(64²⁵, 684, F₆₄, 12, 12) (dual of [(684, 12), 8183, 13]-NRT-code), using
    - OA 6-folding and stacking [i] based on linear OA(64²⁵, 4104, F₆₄, 12) (dual of [4104, 4079, 13]-code), using
      - constructionÂ X applied to C_e(11) âŠ‚ C_e(8) [i] based on
        linear OA(64²³, 4096, F₆₄, 12) (dual of [4096, 4073, 13]-code), using an extension C_e(11) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]
        linear OA(64¹⁷, 4096, F₆₄, 9) (dual of [4096, 4079, 10]-code), using an extension C_e(8) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,8], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]
        linear OA(64², 8, F₆₄, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,64)), using
        discarding factors / shortening the dual code based on linear OA(64², 64, F₆₄, 2) (dual of [64, 62, 3]-code or 64-arc in PG(1,64)), using
        Reedâ€“Solomon code RS(62,64) [i]

(19, 31, 2769)-Net over F₃₂ — Digital

Digital (19, 31, 2769)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(19, 31, 5773888)-Net in Base 32 — Upper bound on s

There is no (19, 31, 5773889)-net in base 32, because

the generalized Rao bound for nets shows that 32^mÂ â‰¥ 45671 940626 851276 678304 909713 758709 977711 218080 > 32³¹ [i]