MinT - Best Known (19, 19+19, s)-Nets in Base 32

Best Known (19, 19+19, s)-Nets in Base 32

(19, 19+19, 162)-Net over F₃₂ — Constructive and digital

Digital (19, 38, 162)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (3, 12, 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, 26, 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]

(19, 19+19, 261)-Net in Base 32 — Constructive

(19, 38, 261)-net in base 32, using

2 times m-reduction [i] based on (19, 40, 261)-net in base 32, using
- base change [i] based on digital (4, 25, 261)-net over F₂₅₆, using
  - net from sequence [i] based on digital (4, 260)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(19, 19+19, 515)-Net over F₃₂ — Digital

Digital (19, 38, 515)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(32³⁸, 515, F₃₂, 2, 19) (dual of [(515, 2), 992, 20]-NRT-code), using
- OOA 2-folding [i] based on linear OA(32³⁸, 1030, F₃₂, 19) (dual of [1030, 992, 20]-code), using
  - constructionÂ X applied to C([0,9]) âŠ‚ C([0,8]) [i] based on
    1. linear OA(32³⁷, 1025, F₃₂, 19) (dual of [1025, 988, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025Â | 32⁴âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (BCH-bound) [i]
    2. linear OA(32³³, 1025, F₃₂, 17) (dual of [1025, 992, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025Â | 32⁴âˆ’1, defining interval IÂ =Â [0,8], and minimum distance dÂ â‰¥ |{−8,−7,…,8}|+1Â =Â 18 (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]

(19, 19+19, 206167)-Net in Base 32 — Upper bound on s

There is no (19, 38, 206168)-net in base 32, because

1 times m-reduction [i] would yield (19, 37, 206168)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 49 041627 030708 093217 631051 519541 648092 661446 667269 047810 > 32³⁷ [i]