MinT - Best Known (19, 23, s)-Nets in Base 32

Best Known (19, 23, s)-Nets in Base 32

(19, 23, large)-Net over F₃₂ — Constructive and digital

Digital (19, 23, large)-net over F₃₂, using

32¹ times duplication [i] based on digital (18, 22, large)-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 (16, 20, 8388602)-net over F₃₂, using
    - net defined by OOA [i] based on linear OOA(32²⁰, 8388602, F₃₂, 6, 4) (dual of [(8388602, 6), 50331592, 5]-NRT-code), using
      - OOA stacking with additional row [i] based on linear OOA(32²⁰, large, F₃₂, 2, 4), using
        generalized (u,Â u+v)-construction [i] based on
        linear OOA(32⁰, 524290, F₃₂, 2, 0) (dual of [(524290, 2), 1048580, 1]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(32⁰, s, F₃₂, 2, 0) with arbitrarily large s, using
        OOA with only one run / dual of NRT-code without redundancy [i]
        linear OOA(32⁰, 524290, F₃₂, 2, 0) (dual of [(524290, 2), 1048580, 1]-NRT-code) (see above)
        linear OOA(32⁰, 524290, F₃₂, 2, 0) (dual of [(524290, 2), 1048580, 1]-NRT-code) (see above)
        linear OOA(32⁰, 524290, F₃₂, 2, 0) (dual of [(524290, 2), 1048580, 1]-NRT-code) (see above)
        linear OOA(32⁰, 524290, F₃₂, 2, 0) (dual of [(524290, 2), 1048580, 1]-NRT-code) (see above)
        linear OOA(32⁰, 524290, F₃₂, 2, 0) (dual of [(524290, 2), 1048580, 1]-NRT-code) (see above)
        linear OOA(32⁰, 524290, F₃₂, 2, 0) (dual of [(524290, 2), 1048580, 1]-NRT-code) (see above)
        linear OOA(32⁰, 524290, F₃₂, 2, 0) (dual of [(524290, 2), 1048580, 1]-NRT-code) (see above)
        linear OOA(32⁰, 524290, F₃₂, 2, 0) (dual of [(524290, 2), 1048580, 1]-NRT-code) (see above)
        linear OOA(32⁰, 524290, F₃₂, 2, 0) (dual of [(524290, 2), 1048580, 1]-NRT-code) (see above)
        linear OOA(32⁰, 524290, F₃₂, 2, 0) (dual of [(524290, 2), 1048580, 1]-NRT-code) (see above)
        linear OOA(32⁰, 524290, F₃₂, 2, 0) (dual of [(524290, 2), 1048580, 1]-NRT-code) (see above)
        linear OOA(32⁰, 524290, F₃₂, 2, 0) (dual of [(524290, 2), 1048580, 1]-NRT-code) (see above)
        linear OOA(32⁰, 524290, F₃₂, 2, 0) (dual of [(524290, 2), 1048580, 1]-NRT-code) (see above)
        linear OOA(32⁰, 524290, F₃₂, 2, 0) (dual of [(524290, 2), 1048580, 1]-NRT-code) (see above)
        linear OOA(32⁰, 524290, F₃₂, 2, 0) (dual of [(524290, 2), 1048580, 1]-NRT-code) (see above)
        linear OOA(32⁰, 524290, F₃₂, 2, 0) (dual of [(524290, 2), 1048580, 1]-NRT-code) (see above)
        linear OOA(32⁰, 524290, F₃₂, 2, 0) (dual of [(524290, 2), 1048580, 1]-NRT-code) (see above)
        linear OOA(32⁰, 524290, F₃₂, 2, 0) (dual of [(524290, 2), 1048580, 1]-NRT-code) (see above)
        linear OOA(32⁰, 524290, F₃₂, 2, 0) (dual of [(524290, 2), 1048580, 1]-NRT-code) (see above)
        linear OOA(32⁰, 524290, F₃₂, 2, 0) (dual of [(524290, 2), 1048580, 1]-NRT-code) (see above)
        linear OOA(32⁰, 524290, F₃₂, 2, 0) (dual of [(524290, 2), 1048580, 1]-NRT-code) (see above)
        linear OOA(32⁰, 524290, F₃₂, 2, 0) (dual of [(524290, 2), 1048580, 1]-NRT-code) (see above)
        linear OOA(32⁰, 524290, F₃₂, 2, 0) (dual of [(524290, 2), 1048580, 1]-NRT-code) (see above)
        linear OOA(32⁰, 524290, F₃₂, 2, 0) (dual of [(524290, 2), 1048580, 1]-NRT-code) (see above)
        linear OOA(32⁰, 524290, F₃₂, 2, 0) (dual of [(524290, 2), 1048580, 1]-NRT-code) (see above)
        linear OOA(32⁰, 524290, F₃₂, 2, 0) (dual of [(524290, 2), 1048580, 1]-NRT-code) (see above)
        linear OOA(32⁰, 524290, F₃₂, 2, 0) (dual of [(524290, 2), 1048580, 1]-NRT-code) (see above)
        linear OOA(32¹, 524290, F₃₂, 2, 1) (dual of [(524290, 2), 1048579, 2]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(32¹, s, F₃₂, 2, 1) with arbitrarily large s, using
        appending 1 arbitrary column [i] based on linear OA(32¹, s, F₃₂, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]
        linear OOA(32¹, 524290, F₃₂, 2, 1) (dual of [(524290, 2), 1048579, 2]-NRT-code) (see above)
        linear OOA(32⁵, 524290, F₃₂, 2, 2) (dual of [(524290, 2), 1048575, 3]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(32⁵, 1082401, F₃₂, 2, 2) (dual of [(1082401, 2), 2164797, 3]-NRT-code), using
        appending kth column [i] based on linear OA(32⁵, 1082401, F₃₂, 2) (dual of [1082401, 1082396, 3]-code), using
        Hamming code H(5,32) [i]
        linear OOA(32¹³, 524290, F₃₂, 2, 4) (dual of [(524290, 2), 1048567, 5]-NRT-code), using
        OOA 2-folding [i] based on linear OA(32¹³, 1048580, F₃₂, 4) (dual of [1048580, 1048567, 5]-code), using
        constructionÂ X applied to C_e(3) âŠ‚ C_e(2) [i] based on
        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(32⁹, 1048576, F₃₂, 3) (dual of [1048576, 1048567, 4]-code or 1048576-cap in PG(8,32)), using an extension C_e(2) of the primitive narrow-sense BCH-code C(I) with length 1048575Â = 32⁴âˆ’1, defining interval IÂ =Â [1,2], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 3 [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]

(19, 23, large)-Net in Base 32 — Upper bound on s

There is no (19, 23, large)-net in base 32, because

2 times m-reduction [i] would yield (19, 21, large)-net in base 32, but
- mutually orthogonal hypercube bound [i]

Best Known (19, 23, s)-Nets in Base 32

(19, 23, large)-Net over F32 — Constructive and digital

(19, 23, large)-Net in Base 32 — Upper bound on s

(19, 23, large)-Net over F₃₂ — Constructive and digital