MinT - Best Known (17, 17+16, s)-Nets in Base 32

Best Known (17, 17+16, s)-Nets in Base 32

(17, 17+16, 142)-Net over F₃₂ — Constructive and digital

Digital (17, 33, 142)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (4, 12, 66)-net over F₃₂, using
  - (u,Â u+v)-construction [i] based on
    1. digital (0, 4, 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]
    2. digital (0, 8, 33)-net over F₃₂, using
      - net from sequence [i] based on digital (0, 32)-sequence over F₃₂ (see above)
2. digital (5, 21, 76)-net over F₃₂, using
  - net from sequence [i] based on digital (5, 75)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 5 and N(F) ≥ 76, using
      - a function field by SÃ©mirat [i]

(17, 17+16, 261)-Net in Base 32 — Constructive

(17, 33, 261)-net in base 32, using

32¹ times duplication [i] based on (16, 32, 261)-net in base 32, using
- base change [i] based on digital (4, 20, 261)-net over F₂₅₆, using
  - net from sequence [i] based on digital (4, 260)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(17, 17+16, 531)-Net over F₃₂ — Digital

Digital (17, 33, 531)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32³³, 531, F₃₂, 16) (dual of [531, 498, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(32³³, 1032, F₃₂, 16) (dual of [1032, 999, 17]-code), using
  - constructionÂ X applied to C_e(15) âŠ‚ C_e(12) [i] based on
    1. linear OA(32³¹, 1024, F₃₂, 16) (dual of [1024, 993, 17]-code), using an extension C_e(15) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [1,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]
    2. linear OA(32²⁵, 1024, F₃₂, 13) (dual of [1024, 999, 14]-code), using an extension C_e(12) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]
    3. linear OA(32², 8, F₃₂, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,32)), using
      - discarding factors / shortening the dual code based on linear OA(32², 32, F₃₂, 2) (dual of [32, 30, 3]-code or 32-arc in PG(1,32)), using
        Reedâ€“Solomon code RS(30,32) [i]

(17, 17+16, 196364)-Net in Base 32 — Upper bound on s

There is no (17, 33, 196365)-net in base 32, because

the generalized Rao bound for nets shows that 32^mÂ â‰¥ 46 768363 593483 096362 319442 613856 850068 194392 844239 > 32³³ [i]