MinT - Best Known (40, 40+37, s)-Nets in Base 32

Best Known (40, 40+37, s)-Nets in Base 32

(40, 40+37, 240)-Net over F₃₂ — Constructive and digital

Digital (40, 77, 240)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (11, 29, 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]
2. digital (11, 48, 120)-net over F₃₂, using
  - net from sequence [i] based on digital (11, 119)-sequence over F₃₂ (see above)

(40, 40+37, 288)-Net in Base 32 — Constructive

(40, 77, 288)-net in base 32, using

31 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]

(40, 40+37, 816)-Net over F₃₂ — Digital

Digital (40, 77, 816)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32⁷⁷, 816, F₃₂, 37) (dual of [816, 739, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(32⁷⁷, 1044, F₃₂, 37) (dual of [1044, 967, 38]-code), using
  - constructionÂ X applied to C_e(36) âŠ‚ C_e(28) [i] based on
    1. linear OA(32⁷⁰, 1024, F₃₂, 37) (dual of [1024, 954, 38]-code), using an extension C_e(36) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [1,36], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 37 [i]
    2. linear OA(32⁵⁷, 1024, F₃₂, 29) (dual of [1024, 967, 30]-code), using an extension C_e(28) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [1,28], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [i]
    3. linear OA(32⁷, 20, F₃₂, 7) (dual of [20, 13, 8]-code or 20-arc in PG(6,32)), using
      - discarding factors / shortening the dual code based on linear OA(32⁷, 32, F₃₂, 7) (dual of [32, 25, 8]-code or 32-arc in PG(6,32)), using
        Reedâ€“Solomon code RS(25,32) [i]

(40, 40+37, 551872)-Net in Base 32 — Upper bound on s

There is no (40, 77, 551873)-net in base 32, because

1 times m-reduction [i] would yield (40, 76, 551873)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 2 462650 275600 898778 163717 227121 534626 670866 865478 762181 939788 337958 091447 692078 020970 621240 945153 616129 327972 459168 > 32⁷⁶ [i]