MinT - Best Known (35, 35+39, s)-Nets in Base 32

Best Known (35, 35+39, s)-Nets in Base 32

(35, 35+39, 202)-Net over F₃₂ — Constructive and digital

Digital (35, 74, 202)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (7, 26, 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, 48, 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]

(35, 35+39, 288)-Net in Base 32 — Constructive

(35, 74, 288)-net in base 32, using

17 times m-reduction [i] based on (35, 91, 288)-net in base 32, using
- base change [i] based on digital (9, 65, 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]

(35, 35+39, 454)-Net over F₃₂ — Digital

Digital (35, 74, 454)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(32⁷⁴, 454, F₃₂, 2, 39) (dual of [(454, 2), 834, 40]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(32⁷⁴, 513, F₃₂, 2, 39) (dual of [(513, 2), 952, 40]-NRT-code), using
  - OOA 2-folding [i] based on linear OA(32⁷⁴, 1026, F₃₂, 39) (dual of [1026, 952, 40]-code), using
    - constructionÂ X applied to C_e(38) âŠ‚ C_e(37) [i] based on
      1. linear OA(32⁷⁴, 1024, F₃₂, 39) (dual of [1024, 950, 40]-code), using an extension C_e(38) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [1,38], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 39 [i]
      2. linear OA(32⁷², 1024, F₃₂, 38) (dual of [1024, 952, 39]-code), using an extension C_e(37) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [1,37], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 38 [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]

(35, 35+39, 155157)-Net in Base 32 — Upper bound on s

There is no (35, 74, 155158)-net in base 32, because

1 times m-reduction [i] would yield (35, 73, 155158)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 75 161952 887019 877828 323796 306257 034865 719663 098774 945235 295457 670481 091117 162227 961757 293195 794628 966439 239790 > 32⁷³ [i]