MinT - Best Known (82−12, 82, s)-Nets in Base 32

Best Known (82−12, 82, s)-Nets in Base 32

(82−12, 82, 2796200)-Net over F₃₂ — Constructive and digital

Digital (70, 82, 2796200)-net over F₃₂, using

net defined by OOA [i] based on linear OOA(32⁸², 2796200, F₃₂, 14, 12) (dual of [(2796200, 14), 39146718, 13]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(32⁸², 8388601, F₃₂, 2, 12) (dual of [(8388601, 2), 16777120, 13]-NRT-code), using
  - discarding factors / shortening the dual code based on linear OOA(32⁸², 8388602, F₃₂, 2, 12) (dual of [(8388602, 2), 16777122, 13]-NRT-code), using
    - (u,Â u+v)-construction [i] based on
      1. linear OOA(32²⁶, 4194301, F₃₂, 2, 6) (dual of [(4194301, 2), 8388576, 7]-NRT-code), using
        OOA 2-folding [i] based on linear OA(32²⁶, 8388602, F₃₂, 6) (dual of [8388602, 8388576, 7]-code), using
        discarding factors / shortening the dual code based on linear OA(32²⁶, large, F₃₂, 6) (dual of [large, large−26, 7]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 33554431Â = 32⁵âˆ’1, defining interval IÂ =Â [0,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
      2. linear OOA(32⁵⁶, 4194301, F₃₂, 2, 12) (dual of [(4194301, 2), 8388546, 13]-NRT-code), using
        OOA 2-folding [i] based on linear OA(32⁵⁶, 8388602, F₃₂, 12) (dual of [8388602, 8388546, 13]-code), using
        discarding factors / shortening the dual code based on linear OA(32⁵⁶, large, F₃₂, 12) (dual of [large, large−56, 13]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 33554431Â = 32⁵âˆ’1, defining interval IÂ =Â [0,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

(82−12, 82, large)-Net over F₃₂ — Digital

Digital (70, 82, large)-net over F₃₂, using

t-expansion [i] based on digital (68, 82, large)-net over F₃₂, using
- 4 times m-reduction [i] based on digital (68, 86, large)-net over F₃₂, using
  - Gilbertâ€“VarÅ¡amov bound [i]

(82−12, 82, large)-Net in Base 32 — Upper bound on s

There is no (70, 82, large)-net in base 32, because

10 times m-reduction [i] would yield (70, 72, large)-net in base 32, but
- mutually orthogonal hypercube bound [i]

Best Known (82−12, 82, s)-Nets in Base 32

(82−12, 82, 2796200)-Net over F32 — Constructive and digital

(82−12, 82, large)-Net over F32 — Digital

(82−12, 82, large)-Net in Base 32 — Upper bound on s

(82−12, 82, 2796200)-Net over F₃₂ — Constructive and digital

(82−12, 82, large)-Net over F₃₂ — Digital