MinT - Best Known (49, 64, s)-Nets in Base 32

Best Known (49, 64, s)-Nets in Base 32

(49, 64, 149830)-Net over F₃₂ — Constructive and digital

Digital (49, 64, 149830)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (0, 7, 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 (42, 57, 149797)-net over F₃₂, using
  - net defined by OOA [i] based on linear OOA(32⁵⁷, 149797, F₃₂, 15, 15) (dual of [(149797, 15), 2246898, 16]-NRT-code), using
    - OOA 7-folding and stacking with additional row [i] based on linear OA(32⁵⁷, 1048580, F₃₂, 15) (dual of [1048580, 1048523, 16]-code), using
      - constructionÂ X applied to C_e(14) âŠ‚ C_e(13) [i] based on
        linear OA(32⁵⁷, 1048576, F₃₂, 15) (dual of [1048576, 1048519, 16]-code), using an extension C_e(14) of the primitive narrow-sense BCH-code C(I) with length 1048575Â = 32⁴âˆ’1, defining interval IÂ =Â [1,14], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]
        linear OA(32⁵³, 1048576, F₃₂, 14) (dual of [1048576, 1048523, 15]-code), using an extension C_e(13) of the primitive narrow-sense BCH-code C(I) with length 1048575Â = 32⁴âˆ’1, defining interval IÂ =Â [1,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [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]

(49, 64, 299594)-Net in Base 32 — Constructive

(49, 64, 299594)-net in base 32, using

32¹ times duplication [i] based on (48, 63, 299594)-net in base 32, using
- base change [i] based on digital (30, 45, 299594)-net over F₁₂₈, using
  - 128¹ times duplication [i] based on digital (29, 44, 299594)-net over F₁₂₈, using
    - net defined by OOA [i] based on linear OOA(128⁴⁴, 299594, F₁₂₈, 15, 15) (dual of [(299594, 15), 4493866, 16]-NRT-code), using
      - OOA 7-folding and stacking with additional row [i] based on linear OA(128⁴⁴, 2097159, F₁₂₈, 15) (dual of [2097159, 2097115, 16]-code), using
        discarding factors / shortening the dual code based on linear OA(128⁴⁴, 2097160, F₁₂₈, 15) (dual of [2097160, 2097116, 16]-code), using
        constructionÂ X applied to C([0,7]) âŠ‚ C([0,6]) [i] based on
        linear OA(128⁴³, 2097153, F₁₂₈, 15) (dual of [2097153, 2097110, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153Â | 128⁶âˆ’1, defining interval IÂ =Â [0,7], and minimum distance dÂ â‰¥ |{−7,−6,…,7}|+1Â =Â 16 (BCH-bound) [i]
        linear OA(128³⁷, 2097153, F₁₂₈, 13) (dual of [2097153, 2097116, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153Â | 128⁶âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]
        linear OA(128¹, 7, F₁₂₈, 1) (dual of [7, 6, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(128¹, s, F₁₂₈, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(49, 64, 1481780)-Net over F₃₂ — Digital

Digital (49, 64, 1481780)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(49, 64, large)-Net in Base 32 — Upper bound on s

There is no (49, 64, large)-net in base 32, because

13 times m-reduction [i] would yield (49, 51, large)-net in base 32, but
- mutually orthogonal hypercube bound [i]

Best Known (49, 64, s)-Nets in Base 32

(49, 64, 149830)-Net over F32 — Constructive and digital

(49, 64, 299594)-Net in Base 32 — Constructive

(49, 64, 1481780)-Net over F32 — Digital

(49, 64, large)-Net in Base 32 — Upper bound on s

(49, 64, 149830)-Net over F₃₂ — Constructive and digital

(49, 64, 1481780)-Net over F₃₂ — Digital