MinT - Best Known (83, 83+17, s)-Nets in Base 32

Best Known (83, 83+17, s)-Nets in Base 32

(83, 83+17, 1048864)-Net over F₃₂ — Constructive and digital

Digital (83, 100, 1048864)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (11, 19, 289)-net over F₃₂, using
  - (u,Â u+v)-construction [i] based on
    1. digital (0, 4, 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 (7, 15, 256)-net over F₃₂, using
      - net defined by OOA [i] based on linear OOA(32¹⁵, 256, F₃₂, 8, 8) (dual of [(256, 8), 2033, 9]-NRT-code), using
        OA 4-folding and stacking [i] based on linear OA(32¹⁵, 1024, F₃₂, 8) (dual of [1024, 1009, 9]-code), using
        an extension C_e(7) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [1,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]
2. digital (64, 81, 1048575)-net over F₃₂, using
  - net defined by OOA [i] based on linear OOA(32⁸¹, 1048575, F₃₂, 17, 17) (dual of [(1048575, 17), 17825694, 18]-NRT-code), using
    - OOA 8-folding and stacking with additional row [i] based on linear OA(32⁸¹, 8388601, F₃₂, 17) (dual of [8388601, 8388520, 18]-code), using
      - discarding factors / shortening the dual code based on linear OA(32⁸¹, large, F₃₂, 17) (dual of [large, large−81, 18]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 33554433Â | 32¹⁰âˆ’1, defining interval IÂ =Â [0,8], and minimum distance dÂ â‰¥ |{−8,−7,…,8}|+1Â =Â 18 (BCH-bound) [i]

(83, 83+17, 1056767)-Net in Base 32 — Constructive

(83, 100, 1056767)-net in base 32, using

(u,Â u+v)-construction [i] based on
1. digital (14, 22, 8192)-net over F₃₂, using
  - net defined by OOA [i] based on linear OOA(32²², 8192, F₃₂, 8, 8) (dual of [(8192, 8), 65514, 9]-NRT-code), using
    - OA 4-folding and stacking [i] based on linear OA(32²², 32768, F₃₂, 8) (dual of [32768, 32746, 9]-code), using
      - an extension C_e(7) of the primitive narrow-sense BCH-code C(I) with length 32767Â = 32³âˆ’1, defining interval IÂ =Â [1,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]
2. (61, 78, 1048575)-net in base 32, using
  - base change [i] based on digital (48, 65, 1048575)-net over F₆₄, using
    - net defined by OOA [i] based on linear OOA(64⁶⁵, 1048575, F₆₄, 17, 17) (dual of [(1048575, 17), 17825710, 18]-NRT-code), using
      - OOA 8-folding and stacking with additional row [i] based on linear OA(64⁶⁵, 8388601, F₆₄, 17) (dual of [8388601, 8388536, 18]-code), using
        discarding factors / shortening the dual code based on linear OA(64⁶⁵, large, F₆₄, 17) (dual of [large, large−65, 18]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 64⁸âˆ’1, defining interval IÂ =Â [0,8], and minimum distance dÂ â‰¥ |{−8,−7,…,8}|+1Â =Â 18 (BCH-bound) [i]

(83, 83+17, large)-Net over F₃₂ — Digital

Digital (83, 100, large)-net over F₃₂, using

5 times m-reduction [i] based on digital (83, 105, large)-net over F₃₂, using
- Gilbertâ€“VarÅ¡amov bound [i]

(83, 83+17, large)-Net in Base 32 — Upper bound on s

There is no (83, 100, large)-net in base 32, because

15 times m-reduction [i] would yield (83, 85, large)-net in base 32, but
- mutually orthogonal hypercube bound [i]

Best Known (83, 83+17, s)-Nets in Base 32

(83, 83+17, 1048864)-Net over F32 — Constructive and digital

(83, 83+17, 1056767)-Net in Base 32 — Constructive

(83, 83+17, large)-Net over F32 — Digital

(83, 83+17, large)-Net in Base 32 — Upper bound on s

(83, 83+17, 1048864)-Net over F₃₂ — Constructive and digital

(83, 83+17, large)-Net over F₃₂ — Digital