MinT - Best Known (108−15, 108, s)-Nets in Base 32

Best Known (108−15, 108, s)-Nets in Base 32

(108−15, 108, 2396786)-Net over F₃₂ — Constructive and digital

Digital (93, 108, 2396786)-net over F₃₂, using

generalized (u,Â u+v)-construction [i] based on
1. digital (1, 6, 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. digital (24, 31, 1198371)-net over F₃₂, using
  - s-reduction based on digital (24, 31, 2796200)-net over F₃₂, using
    - net defined by OOA [i] based on linear OOA(32³¹, 2796200, F₃₂, 7, 7) (dual of [(2796200, 7), 19573369, 8]-NRT-code), using
      - OOA 3-folding and stacking with additional row [i] based on linear OA(32³¹, 8388601, F₃₂, 7) (dual of [8388601, 8388570, 8]-code), using
        discarding factors / shortening the dual code based on linear OA(32³¹, large, F₃₂, 7) (dual of [large, large−31, 8]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 33554433Â | 32¹⁰âˆ’1, defining interval IÂ =Â [0,3], and minimum distance dÂ â‰¥ |{−3,−2,…,3}|+1Â =Â 8 (BCH-bound) [i]
3. digital (56, 71, 1198371)-net over F₃₂, using
  - net defined by OOA [i] based on linear OOA(32⁷¹, 1198371, F₃₂, 15, 15) (dual of [(1198371, 15), 17975494, 16]-NRT-code), using
    - OOA 7-folding and stacking with additional row [i] based on linear OA(32⁷¹, 8388598, F₃₂, 15) (dual of [8388598, 8388527, 16]-code), using
      - discarding factors / shortening the dual code based on linear OA(32⁷¹, large, F₃₂, 15) (dual of [large, large−71, 16]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 33554433Â | 32¹⁰âˆ’1, defining interval IÂ =Â [0,7], and minimum distance dÂ â‰¥ |{−7,−6,…,7}|+1Â =Â 16 (BCH-bound) [i]

(108−15, 108, 2398758)-Net in Base 32 — Constructive

(93, 108, 2398758)-net in base 32, using

base change [i] based on digital (75, 90, 2398758)-net over F₆₄, using
- 64¹ times duplication [i] based on digital (74, 89, 2398758)-net over F₆₄, using
  - generalized (u,Â u+v)-construction [i] based on
    1. digital (2, 7, 2016)-net over F₆₄, using
      - net defined by OOA [i] based on linear OOA(64⁷, 2016, F₆₄, 5, 5) (dual of [(2016, 5), 10073, 6]-NRT-code), using
        OOA 2-folding and stacking with additional row [i] based on linear OA(64⁷, 4033, F₆₄, 5) (dual of [4033, 4026, 6]-code), using
        a code by Danev and Olsson [i]
    2. digital (18, 25, 1198371)-net over F₆₄, using
      - s-reduction based on digital (18, 25, 2796200)-net over F₆₄, using
        net defined by OOA [i] based on linear OOA(64²⁵, 2796200, F₆₄, 7, 7) (dual of [(2796200, 7), 19573375, 8]-NRT-code), using
        OOA 3-folding and stacking with additional row [i] based on linear OA(64²⁵, 8388601, F₆₄, 7) (dual of [8388601, 8388576, 8]-code), using
        discarding factors / shortening the dual code based on linear OA(64²⁵, large, F₆₄, 7) (dual of [large, large−25, 8]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 64⁸âˆ’1, defining interval IÂ =Â [0,3], and minimum distance dÂ â‰¥ |{−3,−2,…,3}|+1Â =Â 8 (BCH-bound) [i]
    3. digital (42, 57, 1198371)-net over F₆₄, using
      - net defined by OOA [i] based on linear OOA(64⁵⁷, 1198371, F₆₄, 15, 15) (dual of [(1198371, 15), 17975508, 16]-NRT-code), using
        OOA 7-folding and stacking with additional row [i] based on linear OA(64⁵⁷, 8388598, F₆₄, 15) (dual of [8388598, 8388541, 16]-code), using
        discarding factors / shortening the dual code based on linear OA(64⁵⁷, large, F₆₄, 15) (dual of [large, large−57, 16]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 64⁸âˆ’1, defining interval IÂ =Â [0,7], and minimum distance dÂ â‰¥ |{−7,−6,…,7}|+1Â =Â 16 (BCH-bound) [i]

(108−15, 108, large)-Net over F₃₂ — Digital

Digital (93, 108, large)-net over F₃₂, using

t-expansion [i] based on digital (87, 108, large)-net over F₃₂, using
- 2 times m-reduction [i] based on digital (87, 110, large)-net over F₃₂, using
  - Gilbertâ€“VarÅ¡amov bound [i]

(108−15, 108, large)-Net in Base 32 — Upper bound on s

There is no (93, 108, large)-net in base 32, because

13 times m-reduction [i] would yield (93, 95, large)-net in base 32, but
- mutually orthogonal hypercube bound [i]

Best Known (108−15, 108, s)-Nets in Base 32

(108−15, 108, 2396786)-Net over F32 — Constructive and digital

(108−15, 108, 2398758)-Net in Base 32 — Constructive

(108−15, 108, large)-Net over F32 — Digital

(108−15, 108, large)-Net in Base 32 — Upper bound on s

(108−15, 108, 2396786)-Net over F₃₂ — Constructive and digital

(108−15, 108, large)-Net over F₃₂ — Digital