MinT - Best Known (51−12, 51, s)-Nets in Base 32

Best Known (51−12, 51, s)-Nets in Base 32

(51−12, 51, 174796)-Net over F₃₂ — Constructive and digital

Digital (39, 51, 174796)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (0, 6, 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 (33, 45, 174763)-net over F₃₂, using
  - net defined by OOA [i] based on linear OOA(32⁴⁵, 174763, F₃₂, 12, 12) (dual of [(174763, 12), 2097111, 13]-NRT-code), using
    - OA 6-folding and stacking [i] based on linear OA(32⁴⁵, 1048578, F₃₂, 12) (dual of [1048578, 1048533, 13]-code), using
      - discarding factors / shortening the dual code based on linear OA(32⁴⁵, 1048580, F₃₂, 12) (dual of [1048580, 1048535, 13]-code), using
        constructionÂ X applied to C_e(11) âŠ‚ C_e(10) [i] based on
        linear OA(32⁴⁵, 1048576, F₃₂, 12) (dual of [1048576, 1048531, 13]-code), using an extension C_e(11) of the primitive narrow-sense BCH-code C(I) with length 1048575Â = 32⁴âˆ’1, defining interval IÂ =Â [1,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]
        linear OA(32⁴¹, 1048576, F₃₂, 11) (dual of [1048576, 1048535, 12]-code), using an extension C_e(10) of the primitive narrow-sense BCH-code C(I) with length 1048575Â = 32⁴âˆ’1, defining interval IÂ =Â [1,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]
        linear OA(32⁰, 4, F₃₂, 0) (dual of [4, 4, 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]

(51−12, 51, 349527)-Net in Base 32 — Constructive

(39, 51, 349527)-net in base 32, using

net defined by OOA [i] based on OOA(32⁵¹, 349527, S₃₂, 12, 12), using
- OA 6-folding and stacking [i] based on OA(32⁵¹, 2097162, S₃₂, 12), using
  - discarding factors based on OA(32⁵¹, 2097163, S₃₂, 12), using
    - discarding parts of the base [i] based on linear OA(128³⁶, 2097163, F₁₂₈, 12) (dual of [2097163, 2097127, 13]-code), using
      - constructionÂ X applied to C_e(11) âŠ‚ C_e(8) [i] based on
        linear OA(128³⁴, 2097152, F₁₂₈, 12) (dual of [2097152, 2097118, 13]-code), using an extension C_e(11) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 128³âˆ’1, defining interval IÂ =Â [1,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]
        linear OA(128²⁵, 2097152, F₁₂₈, 9) (dual of [2097152, 2097127, 10]-code), using an extension C_e(8) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 128³âˆ’1, defining interval IÂ =Â [1,8], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]
        linear OA(128², 11, F₁₂₈, 2) (dual of [11, 9, 3]-code or 11-arc in PG(1,128)), using
        discarding factors / shortening the dual code based on linear OA(128², 128, F₁₂₈, 2) (dual of [128, 126, 3]-code or 128-arc in PG(1,128)), using
        Reedâ€“Solomon code RS(126,128) [i]

(51−12, 51, 1506873)-Net over F₃₂ — Digital

Digital (39, 51, 1506873)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(51−12, 51, large)-Net in Base 32 — Upper bound on s

There is no (39, 51, large)-net in base 32, because

10 times m-reduction [i] would yield (39, 41, large)-net in base 32, but
- mutually orthogonal hypercube bound [i]

Best Known (51−12, 51, s)-Nets in Base 32

(51−12, 51, 174796)-Net over F32 — Constructive and digital

(51−12, 51, 349527)-Net in Base 32 — Constructive

(51−12, 51, 1506873)-Net over F32 — Digital

(51−12, 51, large)-Net in Base 32 — Upper bound on s

(51−12, 51, 174796)-Net over F₃₂ — Constructive and digital

(51−12, 51, 1506873)-Net over F₃₂ — Digital