MinT - Best Known (80, 95, s)-Nets in Base 32

Best Known (80, 95, s)-Nets in Base 32

(80, 95, 1212430)-Net over F₃₂ — Constructive and digital

Digital (80, 95, 1212430)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (17, 24, 14059)-net over F₃₂, using
  - (u,Â u+v)-construction [i] based on
    1. digital (2, 5, 3136)-net over F₃₂, using
      - net defined by OOA [i] based on linear OOA(32⁵, 3136, F₃₂, 3, 3) (dual of [(3136, 3), 9403, 4]-NRT-code), using
        appending kth column [i] based on linear OOA(32⁵, 3136, F₃₂, 2, 3) (dual of [(3136, 2), 6267, 4]-NRT-code), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(32⁵, 3136, F₃₂, 3) (dual of [3136, 3131, 4]-code or 3136-cap in PG(4,32)), using
        caps completed using a computer search [i]
    2. digital (12, 19, 10923)-net over F₃₂, using
      - net defined by OOA [i] based on linear OOA(32¹⁹, 10923, F₃₂, 7, 7) (dual of [(10923, 7), 76442, 8]-NRT-code), using
        OOA 3-folding and stacking with additional row [i] based on linear OA(32¹⁹, 32770, F₃₂, 7) (dual of [32770, 32751, 8]-code), using
        discarding factors / shortening the dual code based on linear OA(32¹⁹, 32771, F₃₂, 7) (dual of [32771, 32752, 8]-code), using
        constructionÂ X applied to C_e(6) âŠ‚ C_e(5) [i] based on
        linear OA(32¹⁹, 32768, F₃₂, 7) (dual of [32768, 32749, 8]-code), using an extension C_e(6) of the primitive narrow-sense BCH-code C(I) with length 32767Â = 32³âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
        linear OA(32¹⁶, 32768, F₃₂, 6) (dual of [32768, 32752, 7]-code), using an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 32767Â = 32³âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
        linear OA(32⁰, 3, F₃₂, 0) (dual of [3, 3, 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]
2. 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]

(80, 95, 1547899)-Net in Base 32 — Constructive

(80, 95, 1547899)-net in base 32, using

(u,Â u+v)-construction [i] based on
1. digital (19, 26, 349528)-net over F₃₂, using
  - net defined by OOA [i] based on linear OOA(32²⁶, 349528, F₃₂, 7, 7) (dual of [(349528, 7), 2446670, 8]-NRT-code), using
    - OOA 3-folding and stacking with additional row [i] based on linear OA(32²⁶, 1048585, F₃₂, 7) (dual of [1048585, 1048559, 8]-code), using
      - discarding factors / shortening the dual code based on linear OA(32²⁶, 1048586, F₃₂, 7) (dual of [1048586, 1048560, 8]-code), using
        constructionÂ X applied to C([0,3]) âŠ‚ C([0,2]) [i] based on
        linear OA(32²⁵, 1048577, F₃₂, 7) (dual of [1048577, 1048552, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577Â | 32⁸âˆ’1, defining interval IÂ =Â [0,3], and minimum distance dÂ â‰¥ |{−3,−2,…,3}|+1Â =Â 8 (BCH-bound) [i]
        linear OA(32¹⁷, 1048577, F₃₂, 5) (dual of [1048577, 1048560, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577Â | 32⁸âˆ’1, defining interval IÂ =Â [0,2], and minimum distance dÂ â‰¥ |{−2,−1,0,1,2}|+1Â =Â 6 (BCH-bound) [i]
        linear OA(32¹, 9, F₃₂, 1) (dual of [9, 8, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(32¹, s, F₃₂, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]
2. (54, 69, 1198371)-net in base 32, using
  - net defined by OOA [i] based on OOA(32⁶⁹, 1198371, S₃₂, 15, 15), using
    - OOA 7-folding and stacking with additional row [i] based on OA(32⁶⁹, 8388598, S₃₂, 15), using
      - discarding factors based on OA(32⁶⁹, large, S₃₂, 15), using
        discarding parts of the base [i] 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]

(80, 95, large)-Net over F₃₂ — Digital

Digital (80, 95, large)-net over F₃₂, using

t-expansion [i] based on digital (79, 95, large)-net over F₃₂, using
- 5 times m-reduction [i] based on digital (79, 100, large)-net over F₃₂, using
  - Gilbertâ€“VarÅ¡amov bound [i]

(80, 95, large)-Net in Base 32 — Upper bound on s

There is no (80, 95, large)-net in base 32, because

13 times m-reduction [i] would yield (80, 82, large)-net in base 32, but
- mutually orthogonal hypercube bound [i]

Best Known (80, 95, s)-Nets in Base 32

(80, 95, 1212430)-Net over F32 — Constructive and digital

(80, 95, 1547899)-Net in Base 32 — Constructive

(80, 95, large)-Net over F32 — Digital

(80, 95, large)-Net in Base 32 — Upper bound on s

(80, 95, 1212430)-Net over F₃₂ — Constructive and digital

(80, 95, large)-Net over F₃₂ — Digital