MinT - Best Known (41, 86, s)-Nets in Base 32

Best Known (41, 86, s)-Nets in Base 32

(41, 86, 218)-Net over F₃₂ — Constructive and digital

Digital (41, 86, 218)-net over F₃₂, using

1 times m-reduction [i] based on digital (41, 87, 218)-net over F₃₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (7, 30, 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]
  2. digital (11, 57, 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]

(41, 86, 288)-Net in Base 32 — Constructive

(41, 86, 288)-net in base 32, using

t-expansion [i] based on (40, 86, 288)-net in base 32, using
- 22 times m-reduction [i] based on (40, 108, 288)-net in base 32, using
  - base change [i] based on (22, 90, 288)-net in base 64, using
    - 1 times m-reduction [i] based on (22, 91, 288)-net in base 64, using
      - base change [i] based on digital (9, 78, 288)-net over F₁₂₈, using
        net from sequence [i] based on digital (9, 287)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 9 and N(F) ≥ 288, using
        a function field by SÃ©mirat [i]

(41, 86, 513)-Net over F₃₂ — Digital

Digital (41, 86, 513)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(32⁸⁶, 513, F₃₂, 2, 45) (dual of [(513, 2), 940, 46]-NRT-code), using
- OOA 2-folding [i] based on linear OA(32⁸⁶, 1026, F₃₂, 45) (dual of [1026, 940, 46]-code), using
  - constructionÂ X applied to C_e(44) âŠ‚ C_e(43) [i] based on
    1. 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]
    2. linear OA(32⁸⁴, 1024, F₃₂, 44) (dual of [1024, 940, 45]-code), using an extension C_e(43) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [1,43], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 44 [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]

(41, 86, 190905)-Net in Base 32 — Upper bound on s

There is no (41, 86, 190906)-net in base 32, because

1 times m-reduction [i] would yield (41, 85, 190906)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 86 652757 752729 669228 070118 289712 963078 151335 647706 311535 497564 786764 562367 250959 813791 257906 253846 294276 673919 123784 622273 033972 > 32⁸⁵ [i]