MinT - Best Known (30, 57, s)-Nets in Base 32

Best Known (30, 57, s)-Nets in Base 32

(30, 57, 202)-Net over F₃₂ — Constructive and digital

Digital (30, 57, 202)-net over F₃₂, using

1 times m-reduction [i] based on digital (30, 58, 202)-net over F₃₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (7, 21, 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 (9, 37, 104)-net over F₃₂, using
    - net from sequence [i] based on digital (9, 103)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 9 and N(F) ≥ 104, using
        a function field by SÃ©mirat [i]

(30, 57, 288)-Net in Base 32 — Constructive

(30, 57, 288)-net in base 32, using

15 times m-reduction [i] based on (30, 72, 288)-net in base 32, using
- base change [i] based on (18, 60, 288)-net in base 64, using
  - 3 times m-reduction [i] based on (18, 63, 288)-net in base 64, using
    - base change [i] based on digital (9, 54, 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]

(30, 57, 761)-Net over F₃₂ — Digital

Digital (30, 57, 761)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32⁵⁷, 761, F₃₂, 27) (dual of [761, 704, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(32⁵⁷, 1038, F₃₂, 27) (dual of [1038, 981, 28]-code), using
  - constructionÂ X applied to C_e(26) âŠ‚ C_e(21) [i] based on
    1. linear OA(32⁵³, 1024, F₃₂, 27) (dual of [1024, 971, 28]-code), using an extension C_e(26) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [1,26], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 27 [i]
    2. linear OA(32⁴³, 1024, F₃₂, 22) (dual of [1024, 981, 23]-code), using an extension C_e(21) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]
    3. linear OA(32⁴, 14, F₃₂, 4) (dual of [14, 10, 5]-code or 14-arc in PG(3,32)), using
      - discarding factors / shortening the dual code based on linear OA(32⁴, 32, F₃₂, 4) (dual of [32, 28, 5]-code or 32-arc in PG(3,32)), using
        Reedâ€“Solomon code RS(28,32) [i]

(30, 57, 556881)-Net in Base 32 — Upper bound on s

There is no (30, 57, 556882)-net in base 32, because

1 times m-reduction [i] would yield (30, 56, 556882)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 1 942710 692703 917167 073235 181742 928679 671315 258045 536871 748806 752338 853767 084567 131940 > 32⁵⁶ [i]