MinT - Best Known (22, 22+23, s)-Nets in Base 32

Best Known (22, 22+23, s)-Nets in Base 32

(22, 22+23, 162)-Net over F₃₂ — Constructive and digital

Digital (22, 45, 162)-net over F₃₂, using

1 times m-reduction [i] based on digital (22, 46, 162)-net over F₃₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (3, 15, 64)-net over F₃₂, using
    - net from sequence [i] based on digital (3, 63)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 3 and N(F) ≥ 64, using
        a function field by SÃ©mirat [i]
  2. digital (7, 31, 98)-net over F₃₂, using
    - net from sequence [i] based on digital (7, 97)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 7 and N(F) ≥ 98, using
        a function field by SÃ©mirat [i]

(22, 22+23, 261)-Net in Base 32 — Constructive

(22, 45, 261)-net in base 32, using

3 times m-reduction [i] based on (22, 48, 261)-net in base 32, using
- base change [i] based on digital (4, 30, 261)-net over F₂₅₆, using
  - net from sequence [i] based on digital (4, 260)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(22, 22+23, 452)-Net over F₃₂ — Digital

Digital (22, 45, 452)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(32⁴⁵, 452, F₃₂, 2, 23) (dual of [(452, 2), 859, 24]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(32⁴⁵, 513, F₃₂, 2, 23) (dual of [(513, 2), 981, 24]-NRT-code), using
  - OOA 2-folding [i] based on linear OA(32⁴⁵, 1026, F₃₂, 23) (dual of [1026, 981, 24]-code), using
    - constructionÂ X applied to C_e(22) âŠ‚ C_e(21) [i] based on
      1. linear OA(32⁴⁵, 1024, F₃₂, 23) (dual of [1024, 979, 24]-code), using an extension C_e(22) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [1,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]
      2. linear OA(32⁴³, 1024, F₃₂, 22) (dual of [1024, 981, 23]-code), using an extension C_e(21) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]
      3. linear OA(32⁰, 2, F₃₂, 0) (dual of [2, 2, 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]

(22, 22+23, 166049)-Net in Base 32 — Upper bound on s

There is no (22, 45, 166050)-net in base 32, because

1 times m-reduction [i] would yield (22, 44, 166050)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 1 685018 400844 914011 882943 860004 053381 222749 681159 259035 836447 656176 > 32⁴⁴ [i]