MinT - Best Known (82−39, 82, s)-Nets in Base 32

Best Known (82−39, 82, s)-Nets in Base 32

(82−39, 82, 240)-Net over F₃₂ — Constructive and digital

Digital (43, 82, 240)-net over F₃₂, using

3 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)

(82−39, 82, 513)-Net in Base 32 — Constructive

(43, 82, 513)-net in base 32, using

8 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]

(82−39, 82, 915)-Net over F₃₂ — Digital

Digital (43, 82, 915)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32⁸², 915, F₃₂, 39) (dual of [915, 833, 40]-code), using
- discarding factors / shortening the dual code based on linear OA(32⁸², 1042, F₃₂, 39) (dual of [1042, 960, 40]-code), using
  - constructionÂ X applied to C([0,19]) âŠ‚ C([0,16]) [i] based on
    1. linear OA(32⁷⁷, 1025, F₃₂, 39) (dual of [1025, 948, 40]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025Â | 32⁴âˆ’1, defining interval IÂ =Â [0,19], and minimum distance dÂ â‰¥ |{−19,−18,…,19}|+1Â =Â 40 (BCH-bound) [i]
    2. linear OA(32⁶⁵, 1025, F₃₂, 33) (dual of [1025, 960, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025Â | 32⁴âˆ’1, defining interval IÂ =Â [0,16], and minimum distance dÂ â‰¥ |{−16,−15,…,16}|+1Â =Â 34 (BCH-bound) [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]

(82−39, 82, 667635)-Net in Base 32 — Upper bound on s

There is no (43, 82, 667636)-net in base 32, because

1 times m-reduction [i] would yield (43, 81, 667636)-net in base 32, but
- 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]