MinT - Best Known (60, 60+35, s)-Nets in Base 32

Best Known (60, 60+35, s)-Nets in Base 32

(60, 60+35, 360)-Net over F₃₂ — Constructive and digital

Digital (60, 95, 360)-net over F₃₂, using

generalized (u,Â u+v)-construction [i] based on
1. digital (10, 21, 120)-net over F₃₂, using
  - s-reduction based on digital (10, 21, 205)-net over F₃₂, using
    - net defined by OOA [i] based on linear OOA(32²¹, 205, F₃₂, 11, 11) (dual of [(205, 11), 2234, 12]-NRT-code), using
      - OOA 5-folding and stacking with additional row [i] based on linear OA(32²¹, 1026, F₃₂, 11) (dual of [1026, 1005, 12]-code), using
        constructionÂ X applied to C_e(10) âŠ‚ C_e(9) [i] based on
        linear OA(32²¹, 1024, F₃₂, 11) (dual of [1024, 1003, 12]-code), using an extension C_e(10) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [1,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]
        linear OA(32¹⁹, 1024, F₃₂, 10) (dual of [1024, 1005, 11]-code), using an extension C_e(9) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [1,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 10 [i]
        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]
2. digital (11, 28, 120)-net over F₃₂, using
  - net from sequence [i] based on digital (11, 119)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 11 and N(F) ≥ 120, using
      - a function field by SÃ©mirat [i]
3. digital (11, 46, 120)-net over F₃₂, using
  - net from sequence [i] based on digital (11, 119)-sequence over F₃₂ (see above)

(60, 60+35, 557)-Net in Base 32 — Constructive

(60, 95, 557)-net in base 32, using

1 times m-reduction [i] based on (60, 96, 557)-net in base 32, using
- (u,Â u+v)-construction [i] based on
  1. digital (1, 19, 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. (41, 77, 513)-net in base 32, using
    - 1 times m-reduction [i] based on (41, 78, 513)-net in base 32, using
      - base change [i] based on digital (28, 65, 513)-net over F₆₄, using
        net from sequence [i] based on digital (28, 512)-sequence over F₆₄, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 28 and N(F) ≥ 513, using
        the Hermitian function field over F₆₄ [i]

(60, 60+35, 7026)-Net over F₃₂ — Digital

Digital (60, 95, 7026)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(60, 60+35, large)-Net in Base 32 — Upper bound on s

There is no (60, 95, large)-net in base 32, because

33 times m-reduction [i] would yield (60, 62, large)-net in base 32, but
- mutually orthogonal hypercube bound [i]

Best Known (60, 60+35, s)-Nets in Base 32

(60, 60+35, 360)-Net over F32 — Constructive and digital

(60, 60+35, 557)-Net in Base 32 — Constructive

(60, 60+35, 7026)-Net over F32 — Digital

(60, 60+35, large)-Net in Base 32 — Upper bound on s

(60, 60+35, 360)-Net over F₃₂ — Constructive and digital

(60, 60+35, 7026)-Net over F₃₂ — Digital