MinT - Best Known (29, 29+30, s)-Nets in Base 32

Best Known (29, 29+30, s)-Nets in Base 32

(29, 29+30, 196)-Net over F₃₂ — Constructive and digital

Digital (29, 59, 196)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (7, 22, 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, 37, 98)-net over F₃₂, using
  - net from sequence [i] based on digital (7, 97)-sequence over F₃₂ (see above)

(29, 29+30, 288)-Net in Base 32 — Constructive

(29, 59, 288)-net in base 32, using

11 times m-reduction [i] based on (29, 70, 288)-net in base 32, using
- base change [i] based on digital (9, 50, 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]

(29, 29+30, 513)-Net over F₃₂ — Digital

Digital (29, 59, 513)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(32⁵⁹, 513, F₃₂, 2, 30) (dual of [(513, 2), 967, 31]-NRT-code), using
- OOA 2-folding [i] based on linear OA(32⁵⁹, 1026, F₃₂, 30) (dual of [1026, 967, 31]-code), using
  - constructionÂ X applied to C_e(29) âŠ‚ C_e(28) [i] based on
    1. linear OA(32⁵⁹, 1024, F₃₂, 30) (dual of [1024, 965, 31]-code), using an extension C_e(29) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [1,29], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 30 [i]
    2. linear OA(32⁵⁷, 1024, F₃₂, 29) (dual of [1024, 967, 30]-code), using an extension C_e(28) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [1,28], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [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]

(29, 29+30, 172441)-Net in Base 32 — Upper bound on s

There is no (29, 59, 172442)-net in base 32, because

the generalized Rao bound for nets shows that 32^mÂ â‰¥ 63658 188659 329244 081435 664304 656884 038396 163154 240335 251830 008976 986610 422550 410928 229288 > 32⁵⁹ [i]