MinT - Best Known (43, 52, s)-Nets in Base 32

Best Known (43, 52, s)-Nets in Base 32

(43, 52, 2113538)-Net over F₃₂ — Constructive and digital

Digital (43, 52, 2113538)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (7, 11, 16388)-net over F₃₂, using
  - net defined by OOA [i] based on linear OOA(32¹¹, 16388, F₃₂, 4, 4) (dual of [(16388, 4), 65541, 5]-NRT-code), using
    - OA 2-folding and stacking [i] based on linear OA(32¹¹, 32776, F₃₂, 4) (dual of [32776, 32765, 5]-code), using
      - constructionÂ X4 applied to C_e(3) âŠ‚ C_e(1) [i] based on
        linear OA(32¹⁰, 32768, F₃₂, 4) (dual of [32768, 32758, 5]-code), using an extension C_e(3) of the primitive narrow-sense BCH-code C(I) with length 32767Â = 32³âˆ’1, defining interval IÂ =Â [1,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]
        linear OA(32⁴, 32768, F₃₂, 2) (dual of [32768, 32764, 3]-code), using an extension C_e(1) of the primitive narrow-sense BCH-code C(I) with length 32767Â = 32³âˆ’1, defining interval IÂ =Â [1,1], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 2 [i]
        linear OA(32⁷, 8, F₃₂, 7) (dual of [8, 1, 8]-code or 8-arc in PG(6,32)), using
        dual of repetition code with length 8 [i]
        linear OA(32¹, 8, F₃₂, 1) (dual of [8, 7, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(32¹, 32, F₃₂, 1) (dual of [32, 31, 2]-code), using
        Reedâ€“Solomon code RS(31,32) [i]
2. digital (32, 41, 2097150)-net over F₃₂, using
  - net defined by OOA [i] based on linear OOA(32⁴¹, 2097150, F₃₂, 9, 9) (dual of [(2097150, 9), 18874309, 10]-NRT-code), using
    - OOA 4-folding and stacking with additional row [i] based on linear OA(32⁴¹, 8388601, F₃₂, 9) (dual of [8388601, 8388560, 10]-code), using
      - discarding factors / shortening the dual code based on linear OA(32⁴¹, large, F₃₂, 9) (dual of [large, large−41, 10]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 33554433Â | 32¹⁰âˆ’1, defining interval IÂ =Â [0,4], and minimum distance dÂ â‰¥ |{−4,−3,…,4}|+1Â =Â 10 (BCH-bound) [i]

(43, 52, 2228223)-Net in Base 32 — Constructive

(43, 52, 2228223)-net in base 32, using

(u,Â u+v)-construction [i] based on
1. (8, 12, 131073)-net in base 32, using
  - base change [i] based on digital (6, 10, 131073)-net over F₆₄, using
    - net defined by OOA [i] based on linear OOA(64¹⁰, 131073, F₆₄, 4, 4) (dual of [(131073, 4), 524282, 5]-NRT-code), using
      - appending kth column [i] based on linear OOA(64¹⁰, 131073, F₆₄, 3, 4) (dual of [(131073, 3), 393209, 5]-NRT-code), using
        OA 2-folding and stacking [i] based on linear OA(64¹⁰, 262146, F₆₄, 4) (dual of [262146, 262136, 5]-code), using
        discarding factors / shortening the dual code based on linear OA(64¹⁰, 262147, F₆₄, 4) (dual of [262147, 262137, 5]-code), using
        constructionÂ X applied to C_e(3) âŠ‚ C_e(2) [i] based on
        linear OA(64¹⁰, 262144, F₆₄, 4) (dual of [262144, 262134, 5]-code), using an extension C_e(3) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 64³âˆ’1, defining interval IÂ =Â [1,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]
        linear OA(64⁷, 262144, F₆₄, 3) (dual of [262144, 262137, 4]-code or 262144-cap in PG(6,64)), using an extension C_e(2) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 64³âˆ’1, defining interval IÂ =Â [1,2], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 3 [i]
        linear OA(64⁰, 3, F₆₄, 0) (dual of [3, 3, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(64⁰, s, F₆₄, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]
2. (31, 40, 2097150)-net in base 32, using
  - base change [i] based on digital (16, 25, 2097150)-net over F₂₅₆, using
    - net defined by OOA [i] based on linear OOA(256²⁵, 2097150, F₂₅₆, 9, 9) (dual of [(2097150, 9), 18874325, 10]-NRT-code), using
      - OOA 4-folding and stacking with additional row [i] based on linear OA(256²⁵, 8388601, F₂₅₆, 9) (dual of [8388601, 8388576, 10]-code), using
        discarding factors / shortening the dual code based on linear OA(256²⁵, large, F₂₅₆, 9) (dual of [large, large−25, 10]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 256³âˆ’1, defining interval IÂ =Â [0,8], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 10 [i]

(43, 52, large)-Net over F₃₂ — Digital

Digital (43, 52, large)-net over F₃₂, using

32¹ times duplication [i] based on digital (42, 51, large)-net over F₃₂, using
- t-expansion [i] based on digital (40, 51, large)-net over F₃₂, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] 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]

(43, 52, large)-Net in Base 32 — Upper bound on s

There is no (43, 52, large)-net in base 32, because

7 times m-reduction [i] would yield (43, 45, large)-net in base 32, but
- mutually orthogonal hypercube bound [i]

Best Known (43, 52, s)-Nets in Base 32

(43, 52, 2113538)-Net over F32 — Constructive and digital

(43, 52, 2228223)-Net in Base 32 — Constructive

(43, 52, large)-Net over F32 — Digital

(43, 52, large)-Net in Base 32 — Upper bound on s

(43, 52, 2113538)-Net over F₃₂ — Constructive and digital

(43, 52, large)-Net over F₃₂ — Digital