MinT - Best Known (43, 81, s)-Nets in Base 32

Best Known (43, 81, s)-Nets in Base 32

(43, 81, 240)-Net over F₃₂ — Constructive and digital

Digital (43, 81, 240)-net over F₃₂, using

4 times m-reduction [i] based on digital (43, 85, 240)-net over F₃₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (11, 32, 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, 53, 120)-net over F₃₂, using
    - net from sequence [i] based on digital (11, 119)-sequence over F₃₂ (see above)

(43, 81, 513)-Net in Base 32 — Constructive

(43, 81, 513)-net in base 32, using

9 times m-reduction [i] based on (43, 90, 513)-net in base 32, using
- base change [i] based on digital (28, 75, 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]

(43, 81, 1002)-Net over F₃₂ — Digital

Digital (43, 81, 1002)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32⁸¹, 1002, F₃₂, 38) (dual of [1002, 921, 39]-code), using
- discarding factors / shortening the dual code based on linear OA(32⁸¹, 1050, F₃₂, 38) (dual of [1050, 969, 39]-code), using
  - constructionÂ X applied to C_e(37) âŠ‚ C_e(27) [i] based on
    1. linear OA(32⁷², 1024, F₃₂, 38) (dual of [1024, 952, 39]-code), using an extension C_e(37) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [1,37], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 38 [i]
    2. linear OA(32⁵⁵, 1024, F₃₂, 28) (dual of [1024, 969, 29]-code), using an extension C_e(27) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [1,27], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 28 [i]
    3. linear OA(32⁹, 26, F₃₂, 9) (dual of [26, 17, 10]-code or 26-arc in PG(8,32)), using
      - discarding factors / shortening the dual code based on linear OA(32⁹, 32, F₃₂, 9) (dual of [32, 23, 10]-code or 32-arc in PG(8,32)), using
        Reedâ€“Solomon code RS(23,32) [i]

(43, 81, 667635)-Net in Base 32 — Upper bound on s

There is no (43, 81, 667636)-net in base 32, because

the generalized Rao bound for nets shows that 32^mÂ â‰¥ 82 632625 837239 432258 581291 264762 371746 509192 708709 305127 740235 623254 405470 331401 426705 996440 827787 495958 828840 374243 931533 > 32⁸¹ [i]