MinT - Best Known (21, 43, s)-Nets in Base 32

Best Known (21, 43, s)-Nets in Base 32

(21, 43, 162)-Net over F₃₂ — Constructive and digital

Digital (21, 43, 162)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (3, 14, 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, 29, 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]

(21, 43, 261)-Net in Base 32 — Constructive

(21, 43, 261)-net in base 32, using

32¹ times duplication [i] based on (20, 42, 261)-net in base 32, using
- base change [i] based on (13, 35, 261)-net in base 64, using
  - 1 times m-reduction [i] based on (13, 36, 261)-net in base 64, using
    - base change [i] based on digital (4, 27, 261)-net over F₂₅₆, using
      - net from sequence [i] based on digital (4, 260)-sequence over F₂₅₆, using
        Niederreiter sequence [i]

(21, 43, 444)-Net over F₃₂ — Digital

Digital (21, 43, 444)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(32⁴³, 444, F₃₂, 2, 22) (dual of [(444, 2), 845, 23]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(32⁴³, 513, F₃₂, 2, 22) (dual of [(513, 2), 983, 23]-NRT-code), using
  - OOA 2-folding [i] based on linear OA(32⁴³, 1026, F₃₂, 22) (dual of [1026, 983, 23]-code), using
    - constructionÂ X applied to C_e(21) âŠ‚ C_e(20) [i] based on
      1. 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]
      2. linear OA(32⁴¹, 1024, F₃₂, 21) (dual of [1024, 983, 22]-code), using an extension C_e(20) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [1,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [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]

(21, 43, 121171)-Net in Base 32 — Upper bound on s

There is no (21, 43, 121172)-net in base 32, because

the generalized Rao bound for nets shows that 32^mÂ â‰¥ 52657 357508 697693 955031 718208 021810 634256 597407 128722 080765 085578 > 32⁴³ [i]