MinT - Best Known (49, 49+11, s)-Nets in Base 32

Best Known (49, 49+11, s)-Nets in Base 32

(49, 49+11, 1678249)-Net over F₃₂ — Constructive and digital

Digital (49, 60, 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 (40, 51, 1677720)-net over F₃₂, using
  - net defined by OOA [i] based on linear OOA(32⁵¹, 1677720, F₃₂, 11, 11) (dual of [(1677720, 11), 18454869, 12]-NRT-code), using
    - OOA 5-folding and stacking with additional row [i] based on linear OA(32⁵¹, 8388601, F₃₂, 11) (dual of [8388601, 8388550, 12]-code), using
      - discarding factors / shortening the dual code based on linear OA(32⁵¹, large, F₃₂, 11) (dual of [large, large−51, 12]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 33554433Â | 32¹⁰âˆ’1, defining interval IÂ =Â [0,5], and minimum distance dÂ â‰¥ |{−5,−4,…,5}|+1Â =Â 12 (BCH-bound) [i]

(49, 49+11, 1685848)-Net in Base 32 — Constructive

(49, 60, 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. (39, 50, 1677720)-net in base 32, using
  - net defined by OOA [i] based on OOA(32⁵⁰, 1677720, S₃₂, 11, 11), using
    - OOA 5-folding and stacking with additional row [i] based on OA(32⁵⁰, 8388601, S₃₂, 11), using
      - discarding factors based on OA(32⁵⁰, large, S₃₂, 11), using
        discarding parts of the base [i] based on linear OA(64⁴¹, large, F₆₄, 11) (dual of [large, large−41, 12]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 64⁸âˆ’1, defining interval IÂ =Â [0,5], and minimum distance dÂ â‰¥ |{−5,−4,…,5}|+1Â =Â 12 (BCH-bound) [i]

(49, 49+11, large)-Net over F₃₂ — Digital

Digital (49, 60, large)-net over F₃₂, using

t-expansion [i] based on digital (48, 60, large)-net over F₃₂, using
- 1 times m-reduction [i] based on digital (48, 61, large)-net over F₃₂, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32⁶¹, large, F₃₂, 13) (dual of [large, large−61, 14]-code), using
    - the expurgated narrow-sense BCH-code C(I) with length 33554433Â | 32¹⁰âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]

(49, 49+11, large)-Net in Base 32 — Upper bound on s

There is no (49, 60, large)-net in base 32, because

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