MinT - Best Known (15, 31, s)-Nets in Base 32

Best Known (15, 31, s)-Nets in Base 32

(15, 31, 131)-Net over F₃₂ — Constructive and digital

Digital (15, 31, 131)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (0, 8, 33)-net over F₃₂, using
  - net from sequence [i] based on digital (0, 32)-sequence over F₃₂, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 0 and N(F) ≥ 33, using
      - the rational function field F₃₂(x) [i]
    - Niederreiter sequence [i]
2. digital (7, 23, 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]

(15, 31, 260)-Net in Base 32 — Constructive

(15, 31, 260)-net in base 32, using

1 times m-reduction [i] based on (15, 32, 260)-net in base 32, using
- base change [i] based on digital (3, 20, 260)-net over F₂₅₆, using
  - net from sequence [i] based on digital (3, 259)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(15, 31, 411)-Net over F₃₂ — Digital

Digital (15, 31, 411)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(32³¹, 411, F₃₂, 2, 16) (dual of [(411, 2), 791, 17]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(32³¹, 513, F₃₂, 2, 16) (dual of [(513, 2), 995, 17]-NRT-code), using
  - OOA 2-folding [i] based on linear OA(32³¹, 1026, F₃₂, 16) (dual of [1026, 995, 17]-code), using
    - constructionÂ X applied to C_e(15) âŠ‚ C_e(14) [i] based on
      1. linear OA(32³¹, 1024, F₃₂, 16) (dual of [1024, 993, 17]-code), using an extension C_e(15) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [1,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]
      2. linear OA(32²⁹, 1024, F₃₂, 15) (dual of [1024, 995, 16]-code), using an extension C_e(14) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [1,14], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [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]

(15, 31, 82559)-Net in Base 32 — Upper bound on s

There is no (15, 31, 82560)-net in base 32, because

the generalized Rao bound for nets shows that 32^mÂ â‰¥ 45676 061176 804675 396653 550902 190643 773918 870033 > 32³¹ [i]