MinT - Best Known (46, 46+41, s)-Nets in Base 32

Best Known (46, 46+41, s)-Nets in Base 32

(46, 46+41, 240)-Net over F₃₂ — Constructive and digital

Digital (46, 87, 240)-net over F₃₂, using

7 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)

(46, 46+41, 513)-Net in Base 32 — Constructive

(46, 87, 513)-net in base 32, using

21 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]

(46, 46+41, 1017)-Net over F₃₂ — Digital

Digital (46, 87, 1017)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32⁸⁷, 1017, F₃₂, 41) (dual of [1017, 930, 42]-code), using
- discarding factors / shortening the dual code based on linear OA(32⁸⁷, 1050, F₃₂, 41) (dual of [1050, 963, 42]-code), using
  - constructionÂ X applied to C_e(40) âŠ‚ C_e(30) [i] based on
    1. linear OA(32⁷⁸, 1024, F₃₂, 41) (dual of [1024, 946, 42]-code), using an extension C_e(40) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [1,40], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 41 [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(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]

(46, 46+41, 794481)-Net in Base 32 — Upper bound on s

There is no (46, 87, 794482)-net in base 32, because

1 times m-reduction [i] would yield (46, 86, 794482)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 2772 690642 502653 653565 826783 930083 839574 240306 847542 384489 619727 851116 412919 467065 091333 545285 131343 013105 650111 401385 868156 615288 > 32⁸⁶ [i]