MinT - Best Known (5, 5+7, s)-Nets in Base 32

Best Known (5, 5+7, s)-Nets in Base 32

(5, 5+7, 99)-Net over F₃₂ — Constructive and digital

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)

(5, 5+7, 110)-Net over F₃₂ — Digital

Digital (5, 12, 110)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32¹², 110, F₃₂, 7) (dual of [110, 98, 8]-code), using
- 14 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (1, 13 times 0) [i] based on linear OA(32¹¹, 95, F₃₂, 7) (dual of [95, 84, 8]-code), using
  - constructionÂ X applied to C_e(6) âŠ‚ C_e(5) [i] based on
    1. linear OA(32¹¹, 94, F₃₂, 7) (dual of [94, 83, 8]-code), using an extension C_e(6) of the narrow-sense BCH-code C(I) with length 93Â | 32²âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
    2. linear OA(32¹⁰, 94, F₃₂, 6) (dual of [94, 84, 7]-code), using an extension C_e(5) of the narrow-sense BCH-code C(I) with length 93Â | 32²âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
    3. linear OA(32⁰, 1, F₃₂, 0) (dual of [1, 1, 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]

(5, 5+7, 257)-Net in Base 32 — Constructive

(5, 12, 257)-net in base 32, using

base change [i] based on (3, 10, 257)-net in base 64, using
- 2 times m-reduction [i] based on (3, 12, 257)-net in base 64, using
  - base change [i] based on digital (0, 9, 257)-net over F₂₅₆, using
    - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
      - Niederreiter sequence [i]

(5, 5+7, 19358)-Net in Base 32 — Upper bound on s

There is no (5, 12, 19359)-net in base 32, because

1 times m-reduction [i] would yield (5, 11, 19359)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 36029 347846 488388 > 32¹¹ [i]