MinT - Best Known (84−40, 84, s)-Nets in Base 32

Best Known (84−40, 84, s)-Nets in Base 32

(84−40, 84, 240)-Net over F₃₂ — Constructive and digital

Digital (44, 84, 240)-net over F₃₂, using

4 times m-reduction [i] based on digital (44, 88, 240)-net over F₃₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (11, 33, 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, 55, 120)-net over F₃₂, using
    - net from sequence [i] based on digital (11, 119)-sequence over F₃₂ (see above)

(84−40, 84, 513)-Net in Base 32 — Constructive

(44, 84, 513)-net in base 32, using

12 times m-reduction [i] based on (44, 96, 513)-net in base 32, using
- base change [i] based on digital (28, 80, 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]

(84−40, 84, 922)-Net over F₃₂ — Digital

Digital (44, 84, 922)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32⁸⁴, 922, F₃₂, 40) (dual of [922, 838, 41]-code), using
- discarding factors / shortening the dual code based on linear OA(32⁸⁴, 1047, F₃₂, 40) (dual of [1047, 963, 41]-code), using
  - constructionÂ X applied to C_e(39) âŠ‚ C_e(30) [i] based on
    1. linear OA(32⁷⁶, 1024, F₃₂, 40) (dual of [1024, 948, 41]-code), using an extension C_e(39) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [1,39], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 40 [i]
    2. linear OA(32⁶¹, 1024, F₃₂, 31) (dual of [1024, 963, 32]-code), using an extension C_e(30) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [1,30], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 31 [i]
    3. linear OA(32⁸, 23, F₃₂, 8) (dual of [23, 15, 9]-code or 23-arc in PG(7,32)), using
      - discarding factors / shortening the dual code based on linear OA(32⁸, 32, F₃₂, 8) (dual of [32, 24, 9]-code or 32-arc in PG(7,32)), using
        Reedâ€“Solomon code RS(24,32) [i]

(84−40, 84, 561780)-Net in Base 32 — Upper bound on s

There is no (44, 84, 561781)-net in base 32, because

the generalized Rao bound for nets shows that 32^mÂ â‰¥ 2 707740 571835 929427 550148 721422 727700 263290 670937 289103 478368 590956 396302 382445 050218 242815 242463 522503 428206 847242 736246 828243 > 32⁸⁴ [i]