MinT - Best Known (19, 39, s)-Nets in Base 32

Best Known (19, 39, s)-Nets in Base 32

(19, 39, 142)-Net over F₃₂ — Constructive and digital

Digital (19, 39, 142)-net over F₃₂, using

2 times m-reduction [i] based on digital (19, 41, 142)-net over F₃₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (1, 12, 44)-net over F₃₂, using
    - net from sequence [i] based on digital (1, 43)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 1 and N(F) ≥ 44, using
        a function field by SÃ©mirat [i]
  2. digital (7, 29, 98)-net over F₃₂, using
    - net from sequence [i] based on digital (7, 97)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 7 and N(F) ≥ 98, using
        a function field by SÃ©mirat [i]

(19, 39, 261)-Net in Base 32 — Constructive

(19, 39, 261)-net in base 32, using

1 times m-reduction [i] based on (19, 40, 261)-net in base 32, using
- base change [i] based on digital (4, 25, 261)-net over F₂₅₆, using
  - net from sequence [i] based on digital (4, 260)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(19, 39, 429)-Net over F₃₂ — Digital

Digital (19, 39, 429)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(32³⁹, 429, F₃₂, 2, 20) (dual of [(429, 2), 819, 21]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(32³⁹, 513, F₃₂, 2, 20) (dual of [(513, 2), 987, 21]-NRT-code), using
  - OOA 2-folding [i] based on linear OA(32³⁹, 1026, F₃₂, 20) (dual of [1026, 987, 21]-code), using
    - constructionÂ X applied to C_e(19) âŠ‚ C_e(18) [i] based on
      1. linear OA(32³⁹, 1024, F₃₂, 20) (dual of [1024, 985, 21]-code), using an extension C_e(19) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [1,19], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [i]
      2. linear OA(32³⁷, 1024, F₃₂, 19) (dual of [1024, 987, 20]-code), using an extension C_e(18) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [1,18], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]
      3. linear OA(32⁰, 2, F₃₂, 0) (dual of [2, 2, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(32⁰, s, F₃₂, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(19, 39, 108312)-Net in Base 32 — Upper bound on s

There is no (19, 39, 108313)-net in base 32, because

the generalized Rao bound for nets shows that 32^mÂ â‰¥ 50217 271859 404759 943221 744125 139538 121834 344304 788772 668328 > 32³⁹ [i]