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

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

(35, 35+35, 218)-Net over F₃₂ — Constructive and digital

Digital (35, 70, 218)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (7, 24, 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, 46, 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]

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

(35, 70, 288)-net in base 32, using

21 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+35, 581)-Net over F₃₂ — Digital

Digital (35, 70, 581)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32⁷⁰, 581, F₃₂, 35) (dual of [581, 511, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(32⁷⁰, 1030, F₃₂, 35) (dual of [1030, 960, 36]-code), using
  - constructionÂ X applied to C([0,17]) âŠ‚ C([0,16]) [i] based on
    1. linear OA(32⁶⁹, 1025, F₃₂, 35) (dual of [1025, 956, 36]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025Â | 32⁴âˆ’1, defining interval IÂ =Â [0,17], and minimum distance dÂ â‰¥ |{−17,−16,…,17}|+1Â =Â 36 (BCH-bound) [i]
    2. linear OA(32⁶⁵, 1025, F₃₂, 33) (dual of [1025, 960, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025Â | 32⁴âˆ’1, defining interval IÂ =Â [0,16], and minimum distance dÂ â‰¥ |{−16,−15,…,16}|+1Â =Â 34 (BCH-bound) [i]
    3. linear OA(32¹, 5, F₃₂, 1) (dual of [5, 4, 2]-code), using
      - discarding factors / shortening the dual code based on linear OA(32¹, s, F₃₂, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(35, 35+35, 297652)-Net in Base 32 — Upper bound on s

There is no (35, 70, 297653)-net in base 32, because

1 times m-reduction [i] would yield (35, 69, 297653)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 71 674851 213140 123899 659000 498886 799038 395796 414971 351200 583726 614056 886442 049751 723357 066717 440468 814804 > 32⁶⁹ [i]