MinT - Best Known (49, 63, s)-Nets in Base 32

Best Known (49, 63, s)-Nets in Base 32

(49, 63, 149863)-Net over F₃₂ — Constructive and digital

Digital (49, 63, 149863)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (3, 10, 66)-net over F₃₂, using
  - (u,Â u+v)-construction [i] based on
    1. digital (0, 3, 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 (0, 7, 33)-net over F₃₂, using
      - net from sequence [i] based on digital (0, 32)-sequence over F₃₂ (see above)
2. digital (39, 53, 149797)-net over F₃₂, using
  - net defined by OOA [i] based on linear OOA(32⁵³, 149797, F₃₂, 14, 14) (dual of [(149797, 14), 2097105, 15]-NRT-code), using
    - OA 7-folding and stacking [i] based on linear OA(32⁵³, 1048579, F₃₂, 14) (dual of [1048579, 1048526, 15]-code), using
      - discarding factors / shortening the dual code based on linear OA(32⁵³, 1048580, F₃₂, 14) (dual of [1048580, 1048527, 15]-code), using
        constructionÂ X applied to C_e(13) âŠ‚ C_e(12) [i] based on
        linear OA(32⁵³, 1048576, F₃₂, 14) (dual of [1048576, 1048523, 15]-code), using an extension C_e(13) of the primitive narrow-sense BCH-code C(I) with length 1048575Â = 32⁴âˆ’1, defining interval IÂ =Â [1,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [i]
        linear OA(32⁴⁹, 1048576, F₃₂, 13) (dual of [1048576, 1048527, 14]-code), using an extension C_e(12) of the primitive narrow-sense BCH-code C(I) with length 1048575Â = 32⁴âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [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]

(49, 63, 299626)-Net in Base 32 — Constructive

(49, 63, 299626)-net in base 32, using

(u,Â u+v)-construction [i] based on
1. digital (0, 7, 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. (42, 56, 299593)-net in base 32, using
  - base change [i] based on digital (26, 40, 299593)-net over F₁₂₈, using
    - net defined by OOA [i] based on linear OOA(128⁴⁰, 299593, F₁₂₈, 14, 14) (dual of [(299593, 14), 4194262, 15]-NRT-code), using
      - OA 7-folding and stacking [i] based on linear OA(128⁴⁰, 2097151, F₁₂₈, 14) (dual of [2097151, 2097111, 15]-code), using
        discarding factors / shortening the dual code based on linear OA(128⁴⁰, 2097152, F₁₂₈, 14) (dual of [2097152, 2097112, 15]-code), using
        an extension C_e(13) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 128³âˆ’1, defining interval IÂ =Â [1,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [i]

(49, 63, 3599429)-Net over F₃₂ — Digital

Digital (49, 63, 3599429)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(49, 63, large)-Net in Base 32 — Upper bound on s

There is no (49, 63, large)-net in base 32, because

12 times m-reduction [i] would yield (49, 51, large)-net in base 32, but
- mutually orthogonal hypercube bound [i]

Best Known (49, 63, s)-Nets in Base 32

(49, 63, 149863)-Net over F32 — Constructive and digital

(49, 63, 299626)-Net in Base 32 — Constructive

(49, 63, 3599429)-Net over F32 — Digital

(49, 63, large)-Net in Base 32 — Upper bound on s

(49, 63, 149863)-Net over F₃₂ — Constructive and digital

(49, 63, 3599429)-Net over F₃₂ — Digital