MinT - Best Known (34, 34+38, s)-Nets in Base 32

Best Known (34, 34+38, s)-Nets in Base 32

(34, 34+38, 196)-Net over F₃₂ — Constructive and digital

Digital (34, 72, 196)-net over F₃₂, using

2 times m-reduction [i] based on digital (34, 74, 196)-net over F₃₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (7, 27, 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]
  2. digital (7, 47, 98)-net over F₃₂, using
    - net from sequence [i] based on digital (7, 97)-sequence over F₃₂ (see above)

(34, 34+38, 288)-Net in Base 32 — Constructive

(34, 72, 288)-net in base 32, using

t-expansion [i] based on (33, 72, 288)-net in base 32, using
- 12 times m-reduction [i] based on (33, 84, 288)-net in base 32, using
  - base change [i] based on digital (9, 60, 288)-net over F₁₂₈, using
    - net from sequence [i] based on digital (9, 287)-sequence over F₁₂₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 9 and N(F) ≥ 288, using
        a function field by SÃ©mirat [i]

(34, 34+38, 442)-Net over F₃₂ — Digital

Digital (34, 72, 442)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(32⁷², 442, F₃₂, 2, 38) (dual of [(442, 2), 812, 39]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(32⁷², 513, F₃₂, 2, 38) (dual of [(513, 2), 954, 39]-NRT-code), using
  - OOA 2-folding [i] based on linear OA(32⁷², 1026, F₃₂, 38) (dual of [1026, 954, 39]-code), using
    - constructionÂ X applied to C_e(37) âŠ‚ C_e(36) [i] based on
      1. linear OA(32⁷², 1024, F₃₂, 38) (dual of [1024, 952, 39]-code), using an extension C_e(37) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [1,37], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 38 [i]
      2. linear OA(32⁷⁰, 1024, F₃₂, 37) (dual of [1024, 954, 38]-code), using an extension C_e(36) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [1,36], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 37 [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]

(34, 34+38, 129284)-Net in Base 32 — Upper bound on s

There is no (34, 72, 129285)-net in base 32, because

the generalized Rao bound for nets shows that 32^mÂ â‰¥ 2 348581 990360 985413 184954 560494 102173 737779 695823 463806 959160 061741 818701 781933 776965 545541 938807 719365 917760 > 32⁷² [i]