MinT - Best Known (27, 27+28, s)-Nets in Base 32

Best Known (27, 27+28, s)-Nets in Base 32

(27, 27+28, 174)-Net over F₃₂ — Constructive and digital

Digital (27, 55, 174)-net over F₃₂, using

2 times m-reduction [i] based on digital (27, 57, 174)-net over F₃₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (5, 20, 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]
  2. digital (7, 37, 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]

(27, 27+28, 288)-Net in Base 32 — Constructive

(27, 55, 288)-net in base 32, using

8 times m-reduction [i] based on (27, 63, 288)-net in base 32, using
- base change [i] based on digital (9, 45, 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]

(27, 27+28, 497)-Net over F₃₂ — Digital

Digital (27, 55, 497)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(32⁵⁵, 497, F₃₂, 2, 28) (dual of [(497, 2), 939, 29]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(32⁵⁵, 513, F₃₂, 2, 28) (dual of [(513, 2), 971, 29]-NRT-code), using
  - OOA 2-folding [i] based on linear OA(32⁵⁵, 1026, F₃₂, 28) (dual of [1026, 971, 29]-code), using
    - constructionÂ X applied to C_e(27) âŠ‚ C_e(26) [i] based on
      1. linear OA(32⁵⁵, 1024, F₃₂, 28) (dual of [1024, 969, 29]-code), using an extension C_e(27) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [1,27], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 28 [i]
      2. 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]
      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]

(27, 27+28, 159649)-Net in Base 32 — Upper bound on s

There is no (27, 55, 159650)-net in base 32, because

the generalized Rao bound for nets shows that 32^mÂ â‰¥ 60713 464977 903260 677059 339477 011332 630371 446504 222144 272966 908203 228824 548478 848176 > 32⁵⁵ [i]