MinT - Best Known (78−14, 78, s)-Nets in Base 32

Best Known (78−14, 78, s)-Nets in Base 32

(78−14, 78, 1198470)-Net over F₃₂ — Constructive and digital

Digital (64, 78, 1198470)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (5, 12, 99)-net over F₃₂, using
  - generalized (u,Â u+v)-construction [i] based on
    1. digital (0, 2, 33)-net over F₃₂, using
      - kÂ =Â 2 construction [i]
    2. digital (0, 3, 33)-net over F₃₂, using
      - net from sequence [i] based on digital (0, 32)-sequence over F₃₂, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 0 and N(F) ≥ 33, using
        the rational function field F₃₂(x) [i]
        Niederreiter sequence [i]
    3. digital (0, 7, 33)-net over F₃₂, using
      - net from sequence [i] based on digital (0, 32)-sequence over F₃₂ (see above)
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]

(78−14, 78, 1198714)-Net in Base 32 — Constructive

(64, 78, 1198714)-net in base 32, using

(u,Â u+v)-construction [i] based on
1. digital (7, 14, 343)-net over F₃₂, using
  - net defined by OOA [i] based on linear OOA(32¹⁴, 343, F₃₂, 7, 7) (dual of [(343, 7), 2387, 8]-NRT-code), using
    - OOA 3-folding and stacking with additional row [i] based on linear OA(32¹⁴, 1030, F₃₂, 7) (dual of [1030, 1016, 8]-code), using
      - constructionÂ X applied to C([0,3]) âŠ‚ C([0,2]) [i] based on
        linear OA(32¹³, 1025, F₃₂, 7) (dual of [1025, 1012, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025Â | 32⁴âˆ’1, defining interval IÂ =Â [0,3], and minimum distance dÂ â‰¥ |{−3,−2,…,3}|+1Â =Â 8 (BCH-bound) [i]
        linear OA(32⁹, 1025, F₃₂, 5) (dual of [1025, 1016, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025Â | 32⁴âˆ’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, 5, F₃₂, 1) (dual of [5, 4, 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. (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]

(78−14, 78, large)-Net over F₃₂ — Digital

Digital (64, 78, large)-net over F₃₂, using

3 times m-reduction [i] based on digital (64, 81, large)-net over F₃₂, using
- Gilbertâ€“VarÅ¡amov bound [i]

(78−14, 78, large)-Net in Base 32 — Upper bound on s

There is no (64, 78, large)-net in base 32, because

12 times m-reduction [i] would yield (64, 66, large)-net in base 32, but
- mutually orthogonal hypercube bound [i]

Best Known (78−14, 78, s)-Nets in Base 32

(78−14, 78, 1198470)-Net over F32 — Constructive and digital

(78−14, 78, 1198714)-Net in Base 32 — Constructive

(78−14, 78, large)-Net over F32 — Digital

(78−14, 78, large)-Net in Base 32 — Upper bound on s

(78−14, 78, 1198470)-Net over F₃₂ — Constructive and digital

(78−14, 78, large)-Net over F₃₂ — Digital