MinT - Best Known (64−19, 64, s)-Nets in Base 32

Best Known (64−19, 64, s)-Nets in Base 32

(64−19, 64, 3674)-Net over F₃₂ — Constructive and digital

Digital (45, 64, 3674)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (0, 9, 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]
2. digital (36, 55, 3641)-net over F₃₂, using
  - net defined by OOA [i] based on linear OOA(32⁵⁵, 3641, F₃₂, 19, 19) (dual of [(3641, 19), 69124, 20]-NRT-code), using
    - OOA 9-folding and stacking with additional row [i] based on linear OA(32⁵⁵, 32770, F₃₂, 19) (dual of [32770, 32715, 20]-code), using
      - discarding factors / shortening the dual code based on linear OA(32⁵⁵, 32771, F₃₂, 19) (dual of [32771, 32716, 20]-code), using
        constructionÂ X applied to C_e(18) âŠ‚ C_e(17) [i] based on
        linear OA(32⁵⁵, 32768, F₃₂, 19) (dual of [32768, 32713, 20]-code), using an extension C_e(18) of the primitive narrow-sense BCH-code C(I) with length 32767Â = 32³âˆ’1, defining interval IÂ =Â [1,18], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]
        linear OA(32⁵², 32768, F₃₂, 18) (dual of [32768, 32716, 19]-code), using an extension C_e(17) of the primitive narrow-sense BCH-code C(I) with length 32767Â = 32³âˆ’1, defining interval IÂ =Â [1,17], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 18 [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]

(64−19, 64, 7283)-Net in Base 32 — Constructive

(45, 64, 7283)-net in base 32, using

base change [i] based on digital (21, 40, 7283)-net over F₂₅₆, using
- net defined by OOA [i] based on linear OOA(256⁴⁰, 7283, F₂₅₆, 19, 19) (dual of [(7283, 19), 138337, 20]-NRT-code), using
  - OOA 9-folding and stacking with additional row [i] based on linear OA(256⁴⁰, 65548, F₂₅₆, 19) (dual of [65548, 65508, 20]-code), using
    - constructionÂ X applied to C([0,9]) âŠ‚ C([0,7]) [i] based on
      1. linear OA(256³⁷, 65537, F₂₅₆, 19) (dual of [65537, 65500, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537Â | 256⁴âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (BCH-bound) [i]
      2. linear OA(256²⁹, 65537, F₂₅₆, 15) (dual of [65537, 65508, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537Â | 256⁴âˆ’1, defining interval IÂ =Â [0,7], and minimum distance dÂ â‰¥ |{−7,−6,…,7}|+1Â =Â 16 (BCH-bound) [i]
      3. linear OA(256³, 11, F₂₅₆, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,256) or 11-cap in PG(2,256)), using
        discarding factors / shortening the dual code based on linear OA(256³, 256, F₂₅₆, 3) (dual of [256, 253, 4]-code or 256-arc in PG(2,256) or 256-cap in PG(2,256)), using
        Reedâ€“Solomon code RS(253,256) [i]

(64−19, 64, 54763)-Net over F₃₂ — Digital

Digital (45, 64, 54763)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(64−19, 64, large)-Net in Base 32 — Upper bound on s

There is no (45, 64, large)-net in base 32, because

17 times m-reduction [i] would yield (45, 47, large)-net in base 32, but
- mutually orthogonal hypercube bound [i]

Best Known (64−19, 64, s)-Nets in Base 32

(64−19, 64, 3674)-Net over F32 — Constructive and digital

(64−19, 64, 7283)-Net in Base 32 — Constructive

(64−19, 64, 54763)-Net over F32 — Digital

(64−19, 64, large)-Net in Base 32 — Upper bound on s

(64−19, 64, 3674)-Net over F₃₂ — Constructive and digital

(64−19, 64, 54763)-Net over F₃₂ — Digital