MinT - Best Known (72−21, 72, s)-Nets in Base 32

Best Known (72−21, 72, s)-Nets in Base 32

(72−21, 72, 3321)-Net over F₃₂ — Constructive and digital

Digital (51, 72, 3321)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (1, 11, 44)-net over F₃₂, using
  - net from sequence [i] based on digital (1, 43)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 1 and N(F) ≥ 44, using
      - a function field by SÃ©mirat [i]
2. digital (40, 61, 3277)-net over F₃₂, using
  - net defined by OOA [i] based on linear OOA(32⁶¹, 3277, F₃₂, 21, 21) (dual of [(3277, 21), 68756, 22]-NRT-code), using
    - OOA 10-folding and stacking with additional row [i] based on linear OA(32⁶¹, 32771, F₃₂, 21) (dual of [32771, 32710, 22]-code), using
      - constructionÂ X applied to C_e(20) âŠ‚ C_e(19) [i] based on
        linear OA(32⁶¹, 32768, F₃₂, 21) (dual of [32768, 32707, 22]-code), using an extension C_e(20) of the primitive narrow-sense BCH-code C(I) with length 32767Â = 32³âˆ’1, defining interval IÂ =Â [1,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]
        linear OA(32⁵⁸, 32768, F₃₂, 20) (dual of [32768, 32710, 21]-code), using an extension C_e(19) of the primitive narrow-sense BCH-code C(I) with length 32767Â = 32³âˆ’1, defining interval IÂ =Â [1,19], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [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]

(72−21, 72, 6554)-Net in Base 32 — Constructive

(51, 72, 6554)-net in base 32, using

base change [i] based on digital (24, 45, 6554)-net over F₂₅₆, using
- 256³ times duplication [i] based on digital (21, 42, 6554)-net over F₂₅₆, using
  - net defined by OOA [i] based on linear OOA(256⁴², 6554, F₂₅₆, 21, 21) (dual of [(6554, 21), 137592, 22]-NRT-code), using
    - OOA 10-folding and stacking with additional row [i] based on linear OA(256⁴², 65541, F₂₅₆, 21) (dual of [65541, 65499, 22]-code), using
      - discarding factors / shortening the dual code based on linear OA(256⁴², 65542, F₂₅₆, 21) (dual of [65542, 65500, 22]-code), using
        constructionÂ X applied to C([0,10]) âŠ‚ C([0,9]) [i] based on
        linear OA(256⁴¹, 65537, F₂₅₆, 21) (dual of [65537, 65496, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537Â | 256⁴âˆ’1, defining interval IÂ =Â [0,10], and minimum distance dÂ â‰¥ |{−10,−9,…,10}|+1Â =Â 22 (BCH-bound) [i]
        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]
        linear OA(256¹, 5, F₂₅₆, 1) (dual of [5, 4, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(256¹, s, F₂₅₆, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(72−21, 72, 70234)-Net over F₃₂ — Digital

Digital (51, 72, 70234)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(72−21, 72, large)-Net in Base 32 — Upper bound on s

There is no (51, 72, large)-net in base 32, because

19 times m-reduction [i] would yield (51, 53, large)-net in base 32, but
- mutually orthogonal hypercube bound [i]

Best Known (72−21, 72, s)-Nets in Base 32

(72−21, 72, 3321)-Net over F32 — Constructive and digital

(72−21, 72, 6554)-Net in Base 32 — Constructive

(72−21, 72, 70234)-Net over F32 — Digital

(72−21, 72, large)-Net in Base 32 — Upper bound on s

(72−21, 72, 3321)-Net over F₃₂ — Constructive and digital

(72−21, 72, 70234)-Net over F₃₂ — Digital