MinT - Best Known (91−45, 91, s)-Nets in Base 32

Best Known (91−45, 91, s)-Nets in Base 32

(91−45, 91, 240)-Net over F₃₂ — Constructive and digital

Digital (46, 91, 240)-net over F₃₂, using

3 times m-reduction [i] based on digital (46, 94, 240)-net over F₃₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (11, 35, 120)-net over F₃₂, using
    - net from sequence [i] based on digital (11, 119)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 11 and N(F) ≥ 120, using
        a function field by SÃ©mirat [i]
  2. digital (11, 59, 120)-net over F₃₂, using
    - net from sequence [i] based on digital (11, 119)-sequence over F₃₂ (see above)

(91−45, 91, 513)-Net in Base 32 — Constructive

(46, 91, 513)-net in base 32, using

17 times m-reduction [i] based on (46, 108, 513)-net in base 32, using
- base change [i] based on digital (28, 90, 513)-net over F₆₄, using
  - net from sequence [i] based on digital (28, 512)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 28 and N(F) ≥ 513, using
      - the Hermitian function field over F₆₄ [i]

(91−45, 91, 750)-Net over F₃₂ — Digital

Digital (46, 91, 750)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32⁹¹, 750, F₃₂, 45) (dual of [750, 659, 46]-code), using
- discarding factors / shortening the dual code based on linear OA(32⁹¹, 1041, F₃₂, 45) (dual of [1041, 950, 46]-code), using
  - constructionÂ X applied to C_e(44) âŠ‚ C_e(38) [i] based on
    1. linear OA(32⁸⁶, 1024, F₃₂, 45) (dual of [1024, 938, 46]-code), using an extension C_e(44) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [1,44], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 45 [i]
    2. linear OA(32⁷⁴, 1024, F₃₂, 39) (dual of [1024, 950, 40]-code), using an extension C_e(38) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [1,38], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 39 [i]
    3. linear OA(32⁵, 17, F₃₂, 5) (dual of [17, 12, 6]-code or 17-arc in PG(4,32)), using
      - discarding factors / shortening the dual code based on linear OA(32⁵, 32, F₃₂, 5) (dual of [32, 27, 6]-code or 32-arc in PG(4,32)), using
        Reedâ€“Solomon code RS(27,32) [i]

(91−45, 91, 419673)-Net in Base 32 — Upper bound on s

There is no (46, 91, 419674)-net in base 32, because

1 times m-reduction [i] would yield (46, 90, 419674)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 2907 443508 449008 874898 749594 698442 031013 047834 733997 936283 170525 680197 139888 839734 455485 994719 259069 141912 196345 061084 986535 179047 831404 > 32⁹⁰ [i]