MinT - Best Known (36−16, 36, s)-Nets in Base 32

Best Known (36−16, 36, s)-Nets in Base 32

(36−16, 36, 175)-Net over F₃₂ — Constructive and digital

Digital (20, 36, 175)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (5, 13, 77)-net over F₃₂, using
  - (u,Â u+v)-construction [i] based on
    1. digital (0, 4, 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 (1, 9, 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, 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]

(36−16, 36, 322)-Net in Base 32 — Constructive

(20, 36, 322)-net in base 32, using

base change [i] based on (14, 30, 322)-net in base 64, using
- (u,Â u+v)-construction [i] based on
  1. digital (0, 8, 65)-net over F₆₄, using
    - net from sequence [i] based on digital (0, 64)-sequence over F₆₄, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 0 and N(F) ≥ 65, using
        the rational function field F₆₄(x) [i]
      - Niederreiter sequence [i]
  2. (6, 22, 257)-net in base 64, using
    - 2 times m-reduction [i] based on (6, 24, 257)-net in base 64, using
      - base change [i] based on digital (0, 18, 257)-net over F₂₅₆, using
        net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
        Niederreiter sequence [i]

(36−16, 36, 1078)-Net over F₃₂ — Digital

Digital (20, 36, 1078)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32³⁶, 1078, F₃₂, 16) (dual of [1078, 1042, 17]-code), using
- 46 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (3, 0, 0, 1, 9 times 0, 1, 32 times 0) [i] based on linear OA(32³¹, 1027, F₃₂, 16) (dual of [1027, 996, 17]-code), using
  - constructionÂ XX applied to C₁Â =Â C([1022,13]), C₂Â =Â C([0,14]), C₃Â =Â C₁Â +Â C₂Â =Â C([0,13]), and C_âˆ©Â =Â C₁Â âˆ©Â C₂Â =Â C([1022,14]) [i] based on
    1. linear OA(32²⁹, 1023, F₃₂, 15) (dual of [1023, 994, 16]-code), using the primitive BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â {−1,0,…,13}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]
    2. linear OA(32²⁹, 1023, F₃₂, 15) (dual of [1023, 994, 16]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [0,14], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]
    3. linear OA(32³¹, 1023, F₃₂, 16) (dual of [1023, 992, 17]-code), using the primitive BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â {−1,0,…,14}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]
    4. linear OA(32²⁷, 1023, F₃₂, 14) (dual of [1023, 996, 15]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [0,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]
    5. 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]
    6. linear OA(32⁰, 2, F₃₂, 0) (dual of [2, 2, 1]-code) (see above)

(36−16, 36, 720279)-Net in Base 32 — Upper bound on s

There is no (20, 36, 720280)-net in base 32, because

the generalized Rao bound for nets shows that 32^mÂ â‰¥ 1 532505 358814 264025 490574 743562 413038 232966 646585 004922 > 32³⁶ [i]

Best Known (36−16, 36, s)-Nets in Base 32

(36−16, 36, 175)-Net over F32 — Constructive and digital

(36−16, 36, 322)-Net in Base 32 — Constructive

(36−16, 36, 1078)-Net over F32 — Digital

(36−16, 36, 720279)-Net in Base 32 — Upper bound on s

(36−16, 36, 175)-Net over F₃₂ — Constructive and digital

(36−16, 36, 1078)-Net over F₃₂ — Digital