MinT - Best Known (75, 75+14, s)-Nets in Base 32

Best Known (75, 75+14, s)-Nets in Base 32

(75, 75+14, 1210319)-Net over F₃₂ — Constructive and digital

Digital (75, 89, 1210319)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (16, 23, 11948)-net over F₃₂, using
  - (u,Â u+v)-construction [i] based on
    1. digital (1, 4, 1025)-net over F₃₂, using
      - net defined by OOA [i] based on linear OOA(32⁴, 1025, F₃₂, 3, 3) (dual of [(1025, 3), 3071, 4]-NRT-code), using
        appending kth column [i] based on linear OOA(32⁴, 1025, F₃₂, 2, 3) (dual of [(1025, 2), 2046, 4]-NRT-code), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(32⁴, 1025, F₃₂, 3) (dual of [1025, 1021, 4]-code or 1025-cap in PG(3,32)), using
        ovoid in PG(3,Â 32) [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 (52, 66, 1198371)-net over F₃₂, using
  - net defined by OOA [i] based on linear OOA(32⁶⁶, 1198371, F₃₂, 14, 14) (dual of [(1198371, 14), 16777128, 15]-NRT-code), using
    - OA 7-folding and stacking [i] based on linear OA(32⁶⁶, 8388597, F₃₂, 14) (dual of [8388597, 8388531, 15]-code), using
      - discarding factors / shortening the dual code based on linear OA(32⁶⁶, large, F₃₂, 14) (dual of [large, large−66, 15]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 33554431Â = 32⁵âˆ’1, defining interval IÂ =Â [0,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]

(75, 75+14, 1547897)-Net in Base 32 — Constructive

(75, 89, 1547897)-net in base 32, using

(u,Â u+v)-construction [i] based on
1. digital (18, 25, 349526)-net over F₃₂, using
  - net defined by OOA [i] based on linear OOA(32²⁵, 349526, F₃₂, 7, 7) (dual of [(349526, 7), 2446657, 8]-NRT-code), using
    - OOA 3-folding and stacking with additional row [i] based on linear OA(32²⁵, 1048579, F₃₂, 7) (dual of [1048579, 1048554, 8]-code), using
      - discarding factors / shortening the dual code based on linear OA(32²⁵, 1048580, F₃₂, 7) (dual of [1048580, 1048555, 8]-code), using
        constructionÂ X applied to C_e(6) âŠ‚ C_e(5) [i] based on
        linear OA(32²⁵, 1048576, F₃₂, 7) (dual of [1048576, 1048551, 8]-code), using an extension C_e(6) of the primitive narrow-sense BCH-code C(I) with length 1048575Â = 32⁴âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
        linear OA(32²¹, 1048576, F₃₂, 6) (dual of [1048576, 1048555, 7]-code), using an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 1048575Â = 32⁴âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [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]
2. (50, 64, 1198371)-net in base 32, using
  - base change [i] based on digital (26, 40, 1198371)-net over F₂₅₆, using
    - net defined by OOA [i] based on linear OOA(256⁴⁰, 1198371, F₂₅₆, 14, 14) (dual of [(1198371, 14), 16777154, 15]-NRT-code), using
      - OA 7-folding and stacking [i] based on linear OA(256⁴⁰, 8388597, F₂₅₆, 14) (dual of [8388597, 8388557, 15]-code), using
        discarding factors / shortening the dual code based on linear OA(256⁴⁰, large, F₂₅₆, 14) (dual of [large, large−40, 15]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 256³âˆ’1, defining interval IÂ =Â [0,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]

(75, 75+14, large)-Net over F₃₂ — Digital

Digital (75, 89, large)-net over F₃₂, using

6 times m-reduction [i] based on digital (75, 95, large)-net over F₃₂, using
- Gilbertâ€“VarÅ¡amov bound [i]

(75, 75+14, large)-Net in Base 32 — Upper bound on s

There is no (75, 89, large)-net in base 32, because

12 times m-reduction [i] would yield (75, 77, large)-net in base 32, but
- mutually orthogonal hypercube bound [i]

Best Known (75, 75+14, s)-Nets in Base 32

(75, 75+14, 1210319)-Net over F32 — Constructive and digital

(75, 75+14, 1547897)-Net in Base 32 — Constructive

(75, 75+14, large)-Net over F32 — Digital

(75, 75+14, large)-Net in Base 32 — Upper bound on s

(75, 75+14, 1210319)-Net over F₃₂ — Constructive and digital

(75, 75+14, large)-Net over F₃₂ — Digital