MinT - Best Known (17, 35, s)-Nets in Base 32

Best Known (17, 35, s)-Nets in Base 32

(17, 35, 142)-Net over F₃₂ — Constructive and digital

Digital (17, 35, 142)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (1, 10, 44)-net over F₃₂, using
  - net from sequence [i] based on digital (1, 43)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 1 and N(F) ≥ 44, using
      - a function field by SÃ©mirat [i]
2. digital (7, 25, 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]

(17, 35, 260)-Net in Base 32 — Constructive

(17, 35, 260)-net in base 32, using

1 times m-reduction [i] based on (17, 36, 260)-net in base 32, using
- base change [i] based on (11, 30, 260)-net in base 64, using
  - 2 times m-reduction [i] based on (11, 32, 260)-net in base 64, using
    - base change [i] based on digital (3, 24, 260)-net over F₂₅₆, using
      - net from sequence [i] based on digital (3, 259)-sequence over F₂₅₆, using
        Niederreiter sequence [i]

(17, 35, 417)-Net over F₃₂ — Digital

Digital (17, 35, 417)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(32³⁵, 417, F₃₂, 2, 18) (dual of [(417, 2), 799, 19]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(32³⁵, 513, F₃₂, 2, 18) (dual of [(513, 2), 991, 19]-NRT-code), using
  - OOA 2-folding [i] based on linear OA(32³⁵, 1026, F₃₂, 18) (dual of [1026, 991, 19]-code), using
    - constructionÂ X applied to C_e(17) âŠ‚ C_e(16) [i] based on
      1. linear OA(32³⁵, 1024, F₃₂, 18) (dual of [1024, 989, 19]-code), using an extension C_e(17) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [1,17], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 18 [i]
      2. linear OA(32³³, 1024, F₃₂, 17) (dual of [1024, 991, 18]-code), using an extension C_e(16) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [1,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [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]

(17, 35, 95440)-Net in Base 32 — Upper bound on s

There is no (17, 35, 95441)-net in base 32, because

the generalized Rao bound for nets shows that 32^mÂ â‰¥ 47894 179721 769390 194639 918077 530604 121573 608665 148240 > 32³⁵ [i]