MinT - Best Known (109−6, 109, s)-Nets in Base 32

Best Known (109−6, 109, s)-Nets in Base 32

(109−6, 109, large)-Net over F₃₂ — Constructive and digital

Digital (103, 109, large)-net over F₃₂, using

t-expansion [i] based on digital (99, 109, large)-net over F₃₂, using
- 1 times m-reduction [i] based on digital (99, 110, large)-net over F₃₂, using
  - (u,Â u+v)-construction [i] based on
    1. digital (16, 21, 4194301)-net over F₃₂, using
      - net defined by OOA [i] based on linear OOA(32²¹, 4194301, F₃₂, 5, 5) (dual of [(4194301, 5), 20971484, 6]-NRT-code), using
        OOA 2-folding and stacking with additional row [i] based on linear OA(32²¹, large, F₃₂, 5) (dual of [large, large−21, 6]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 33554433Â | 32¹⁰âˆ’1, defining interval IÂ =Â [0,2], and minimum distance dÂ â‰¥ |{−2,−1,0,1,2}|+1Â =Â 6 (BCH-bound) [i]
    2. digital (78, 89, 4404018)-net over F₃₂, using
      - (u,Â u+v)-construction [i] based on
        digital (16, 21, 4194301)-net over F₃₂ (see above)
        digital (57, 68, 2202009)-net over F₃₂, using
        (u,Â u+v)-construction [i] based on
        digital (12, 17, 524289)-net over F₃₂, using
        net defined by OOA [i] based on linear OOA(32¹⁷, 524289, F₃₂, 5, 5) (dual of [(524289, 5), 2621428, 6]-NRT-code), using
        OOA 2-folding and stacking with additional row [i] based on linear OA(32¹⁷, 1048579, F₃₂, 5) (dual of [1048579, 1048562, 6]-code), using
        discarding factors / shortening the dual code based on linear OA(32¹⁷, 1048580, F₃₂, 5) (dual of [1048580, 1048563, 6]-code), using
        constructionÂ X applied to C_e(4) âŠ‚ C_e(3) [i] based on
        linear OA(32¹⁷, 1048576, F₃₂, 5) (dual of [1048576, 1048559, 6]-code), using an extension C_e(4) of the primitive narrow-sense BCH-code C(I) with length 1048575Â = 32⁴âˆ’1, defining interval IÂ =Â [1,4], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]
        linear OA(32¹³, 1048576, F₃₂, 4) (dual of [1048576, 1048563, 5]-code), using an extension C_e(3) of the primitive narrow-sense BCH-code C(I) with length 1048575Â = 32⁴âˆ’1, defining interval IÂ =Â [1,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]
        linear OA(32⁰, 4, F₃₂, 0) (dual of [4, 4, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(32⁰, s, F₃₂, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]
        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]

(109−6, 109, large)-Net in Base 32 — Upper bound on s

There is no (103, 109, large)-net in base 32, because

4 times m-reduction [i] would yield (103, 105, large)-net in base 32, but
- mutually orthogonal hypercube bound [i]

Best Known (109−6, 109, s)-Nets in Base 32

(109−6, 109, large)-Net over F32 — Constructive and digital

(109−6, 109, large)-Net in Base 32 — Upper bound on s

(109−6, 109, large)-Net over F₃₂ — Constructive and digital