MinT - Best Known (55−10, 55, s)-Nets in Base 32

Best Known (55−10, 55, s)-Nets in Base 32

(55−10, 55, 1678249)-Net over F₃₂ — Constructive and digital

Digital (45, 55, 1678249)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (4, 9, 529)-net over F₃₂, using
  - (u,Â u+v)-construction [i] based on
    1. digital (0, 2, 33)-net over F₃₂, using
      - kÂ =Â 2 construction [i]
    2. digital (2, 7, 496)-net over F₃₂, using
      - net defined by OOA [i] based on linear OOA(32⁷, 496, F₃₂, 5, 5) (dual of [(496, 5), 2473, 6]-NRT-code), using
        OOA 2-folding and stacking with additional row [i] based on linear OA(32⁷, 993, F₃₂, 5) (dual of [993, 986, 6]-code), using
        a code by Danev and Olsson [i]
2. digital (36, 46, 1677720)-net over F₃₂, using
  - net defined by OOA [i] based on linear OOA(32⁴⁶, 1677720, F₃₂, 10, 10) (dual of [(1677720, 10), 16777154, 11]-NRT-code), using
    - OA 5-folding and stacking [i] based on linear OA(32⁴⁶, 8388600, F₃₂, 10) (dual of [8388600, 8388554, 11]-code), using
      - discarding factors / shortening the dual code based on linear OA(32⁴⁶, large, F₃₂, 10) (dual of [large, large−46, 11]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 33554431Â = 32⁵âˆ’1, defining interval IÂ =Â [0,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]

(55−10, 55, 1685848)-Net in Base 32 — Constructive

(45, 55, 1685848)-net in base 32, using

(u,Â u+v)-construction [i] based on
1. (5, 10, 8128)-net in base 32, using
  - net defined by OOA [i] based on OOA(32¹⁰, 8128, S₃₂, 5, 5), using
    - OOA 2-folding and stacking with additional row [i] based on OA(32¹⁰, 16257, S₃₂, 5), using
      - discarding parts of the base [i] based on linear OA(128⁷, 16257, F₁₂₈, 5) (dual of [16257, 16250, 6]-code), using
        a code by Danev and Olsson [i]
2. (35, 45, 1677720)-net in base 32, using
  - net defined by OOA [i] based on OOA(32⁴⁵, 1677720, S₃₂, 10, 10), using
    - OA 5-folding and stacking [i] based on OA(32⁴⁵, 8388600, S₃₂, 10), using
      - discarding factors based on OA(32⁴⁵, large, S₃₂, 10), using
        discarding parts of the base [i] based on linear OA(64³⁷, large, F₆₄, 10) (dual of [large, large−37, 11]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 64⁴âˆ’1, defining interval IÂ =Â [0,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]

(55−10, 55, large)-Net over F₃₂ — Digital

Digital (45, 55, large)-net over F₃₂, using

t-expansion [i] based on digital (44, 55, large)-net over F₃₂, using
- 1 times m-reduction [i] based on digital (44, 56, large)-net over F₃₂, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] 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]

(55−10, 55, large)-Net in Base 32 — Upper bound on s

There is no (45, 55, large)-net in base 32, because

8 times m-reduction [i] would yield (45, 47, large)-net in base 32, but
- mutually orthogonal hypercube bound [i]