MinT - Best Known (27, 27+27, s)-Nets in Base 32

Best Known (27, 27+27, s)-Nets in Base 32

(27, 27+27, 196)-Net over F₃₂ — Constructive and digital

Digital (27, 54, 196)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (7, 20, 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, 34, 98)-net over F₃₂, using
  - net from sequence [i] based on digital (7, 97)-sequence over F₃₂ (see above)

(27, 27+27, 288)-Net in Base 32 — Constructive

(27, 54, 288)-net in base 32, using

9 times m-reduction [i] based on (27, 63, 288)-net in base 32, using
- base change [i] based on digital (9, 45, 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]

(27, 27+27, 515)-Net over F₃₂ — Digital

Digital (27, 54, 515)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(32⁵⁴, 515, F₃₂, 2, 27) (dual of [(515, 2), 976, 28]-NRT-code), using
- OOA 2-folding [i] based on linear OA(32⁵⁴, 1030, F₃₂, 27) (dual of [1030, 976, 28]-code), using
  - constructionÂ X applied to C([0,13]) âŠ‚ C([0,12]) [i] based on
    1. linear OA(32⁵³, 1025, F₃₂, 27) (dual of [1025, 972, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025Â | 32⁴âˆ’1, defining interval IÂ =Â [0,13], and minimum distance dÂ â‰¥ |{−13,−12,…,13}|+1Â =Â 28 (BCH-bound) [i]
    2. linear OA(32⁴⁹, 1025, F₃₂, 25) (dual of [1025, 976, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025Â | 32⁴âˆ’1, defining interval IÂ =Â [0,12], and minimum distance dÂ â‰¥ |{−12,−11,…,12}|+1Â =Â 26 (BCH-bound) [i]
    3. linear OA(32¹, 5, F₃₂, 1) (dual of [5, 4, 2]-code), using
      - discarding factors / shortening the dual code based on linear OA(32¹, s, F₃₂, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(27, 27+27, 250273)-Net in Base 32 — Upper bound on s

There is no (27, 54, 250274)-net in base 32, because

1 times m-reduction [i] would yield (27, 53, 250274)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 59 288582 122350 118508 818677 020112 873620 911583 350592 285475 111214 922628 825344 891448 > 32⁵³ [i]