MinT - Best Known (109−23, 109, s)-Nets in Base 32

Best Known (109−23, 109, s)-Nets in Base 32

(109−23, 109, 95435)-Net over F₃₂ — Constructive and digital

Digital (86, 109, 95435)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (9, 20, 110)-net over F₃₂, using
  - generalized (u,Â u+v)-construction [i] based on
    1. 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]
    2. digital (0, 5, 33)-net over F₃₂, using
      - net from sequence [i] based on digital (0, 32)-sequence over F₃₂ (see above)
    3. digital (1, 12, 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 (66, 89, 95325)-net over F₃₂, using
  - net defined by OOA [i] based on linear OOA(32⁸⁹, 95325, F₃₂, 23, 23) (dual of [(95325, 23), 2192386, 24]-NRT-code), using
    - OOA 11-folding and stacking with additional row [i] based on linear OA(32⁸⁹, 1048576, F₃₂, 23) (dual of [1048576, 1048487, 24]-code), using
      - an extension C_e(22) of the primitive narrow-sense BCH-code C(I) with length 1048575Â = 32⁴âˆ’1, defining interval IÂ =Â [1,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]

(109−23, 109, 762600)-Net in Base 32 — Constructive

(86, 109, 762600)-net in base 32, using

32¹ times duplication [i] based on (85, 108, 762600)-net in base 32, using
- base change [i] based on digital (67, 90, 762600)-net over F₆₄, using
  - 64¹ times duplication [i] based on digital (66, 89, 762600)-net over F₆₄, using
    - net defined by OOA [i] based on linear OOA(64⁸⁹, 762600, F₆₄, 23, 23) (dual of [(762600, 23), 17539711, 24]-NRT-code), using
      - OOA 11-folding and stacking with additional row [i] based on linear OA(64⁸⁹, 8388601, F₆₄, 23) (dual of [8388601, 8388512, 24]-code), using
        discarding factors / shortening the dual code based on linear OA(64⁸⁹, large, F₆₄, 23) (dual of [large, large−89, 24]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 64⁸âˆ’1, defining interval IÂ =Â [0,11], and minimum distance dÂ â‰¥ |{−11,−10,…,11}|+1Â =Â 24 (BCH-bound) [i]

(109−23, 109, 8371936)-Net over F₃₂ — Digital

Digital (86, 109, 8371936)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(109−23, 109, large)-Net in Base 32 — Upper bound on s

There is no (86, 109, large)-net in base 32, because

21 times m-reduction [i] would yield (86, 88, large)-net in base 32, but
- mutually orthogonal hypercube bound [i]

Best Known (109−23, 109, s)-Nets in Base 32

(109−23, 109, 95435)-Net over F32 — Constructive and digital

(109−23, 109, 762600)-Net in Base 32 — Constructive

(109−23, 109, 8371936)-Net over F32 — Digital

(109−23, 109, large)-Net in Base 32 — Upper bound on s

(109−23, 109, 95435)-Net over F₃₂ — Constructive and digital

(109−23, 109, 8371936)-Net over F₃₂ — Digital