MinT - Best Known (37−19, 37, s)-Nets in Base 32

Best Known (37−19, 37, s)-Nets in Base 32

(37−19, 37, 142)-Net over F₃₂ — Constructive and digital

Digital (18, 37, 142)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (4, 13, 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, 9, 33)-net over F₃₂, using
      - net from sequence [i] based on digital (0, 32)-sequence over F₃₂ (see above)
2. digital (5, 24, 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]

(37−19, 37, 260)-Net in Base 32 — Constructive

(18, 37, 260)-net in base 32, using

3 times m-reduction [i] based on (18, 40, 260)-net in base 32, using
- base change [i] based on digital (3, 25, 260)-net over F₂₅₆, using
  - net from sequence [i] based on digital (3, 259)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(37−19, 37, 423)-Net over F₃₂ — Digital

Digital (18, 37, 423)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(32³⁷, 423, F₃₂, 2, 19) (dual of [(423, 2), 809, 20]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(32³⁷, 513, F₃₂, 2, 19) (dual of [(513, 2), 989, 20]-NRT-code), using
  - OOA 2-folding [i] based on linear OA(32³⁷, 1026, F₃₂, 19) (dual of [1026, 989, 20]-code), using
    - constructionÂ X applied to C_e(18) âŠ‚ C_e(17) [i] based on
      1. 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]
      2. linear OA(32³⁵, 1024, F₃₂, 18) (dual of [1024, 989, 19]-code), using an extension C_e(17) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [1,17], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 18 [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]

(37−19, 37, 140273)-Net in Base 32 — Upper bound on s

There is no (18, 37, 140274)-net in base 32, because

1 times m-reduction [i] would yield (18, 36, 140274)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 1 532520 837339 981138 110217 549223 730096 918313 847469 055782 > 32³⁶ [i]