MinT - Best Known (33−10, 33, s)-Nets in Base 32

Best Known (33−10, 33, s)-Nets in Base 32

(33−10, 33, 6587)-Net over F₃₂ — Constructive and digital

Digital (23, 33, 6587)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (0, 5, 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 (18, 28, 6554)-net over F₃₂, using
  - net defined by OOA [i] based on linear OOA(32²⁸, 6554, F₃₂, 10, 10) (dual of [(6554, 10), 65512, 11]-NRT-code), using
    - OA 5-folding and stacking [i] based on linear OA(32²⁸, 32770, F₃₂, 10) (dual of [32770, 32742, 11]-code), using
      - discarding factors / shortening the dual code based on linear OA(32²⁸, 32771, F₃₂, 10) (dual of [32771, 32743, 11]-code), using
        constructionÂ X applied to C_e(9) âŠ‚ C_e(8) [i] based on
        linear OA(32²⁸, 32768, F₃₂, 10) (dual of [32768, 32740, 11]-code), using an extension C_e(9) of the primitive narrow-sense BCH-code C(I) with length 32767Â = 32³âˆ’1, defining interval IÂ =Â [1,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 10 [i]
        linear OA(32²⁵, 32768, F₃₂, 9) (dual of [32768, 32743, 10]-code), using an extension C_e(8) of the primitive narrow-sense BCH-code C(I) with length 32767Â = 32³âˆ’1, defining interval IÂ =Â [1,8], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [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]

(33−10, 33, 13108)-Net in Base 32 — Constructive

(23, 33, 13108)-net in base 32, using

32¹ times duplication [i] based on (22, 32, 13108)-net in base 32, using
- base change [i] based on digital (10, 20, 13108)-net over F₂₅₆, using
  - net defined by OOA [i] based on linear OOA(256²⁰, 13108, F₂₅₆, 10, 10) (dual of [(13108, 10), 131060, 11]-NRT-code), using
    - OA 5-folding and stacking [i] based on linear OA(256²⁰, 65540, F₂₅₆, 10) (dual of [65540, 65520, 11]-code), using
      - discarding factors / shortening the dual code based on linear OA(256²⁰, 65541, F₂₅₆, 10) (dual of [65541, 65521, 11]-code), using
        constructionÂ X applied to C_e(9) âŠ‚ C_e(7) [i] based on
        linear OA(256¹⁹, 65536, F₂₅₆, 10) (dual of [65536, 65517, 11]-code), using an extension C_e(9) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 256²âˆ’1, defining interval IÂ =Â [1,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 10 [i]
        linear OA(256¹⁵, 65536, F₂₅₆, 8) (dual of [65536, 65521, 9]-code), using an extension C_e(7) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 256²âˆ’1, defining interval IÂ =Â [1,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [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]

(33−10, 33, 44189)-Net over F₃₂ — Digital

Digital (23, 33, 44189)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(33−10, 33, large)-Net in Base 32 — Upper bound on s

There is no (23, 33, large)-net in base 32, because

8 times m-reduction [i] would yield (23, 25, large)-net in base 32, but
- mutually orthogonal hypercube bound [i]

Best Known (33−10, 33, s)-Nets in Base 32

(33−10, 33, 6587)-Net over F32 — Constructive and digital

(33−10, 33, 13108)-Net in Base 32 — Constructive

(33−10, 33, 44189)-Net over F32 — Digital

(33−10, 33, large)-Net in Base 32 — Upper bound on s

(33−10, 33, 6587)-Net over F₃₂ — Constructive and digital

(33−10, 33, 44189)-Net over F₃₂ — Digital