MinT - Best Known (45, 89, s)-Nets in Base 32

Best Known (45, 89, s)-Nets in Base 32

(45, 89, 240)-Net over F₃₂ — Constructive and digital

Digital (45, 89, 240)-net over F₃₂, using

2 times m-reduction [i] based on digital (45, 91, 240)-net over F₃₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (11, 34, 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, 57, 120)-net over F₃₂, using
    - net from sequence [i] based on digital (11, 119)-sequence over F₃₂ (see above)

(45, 89, 513)-Net in Base 32 — Constructive

(45, 89, 513)-net in base 32, using

13 times m-reduction [i] based on (45, 102, 513)-net in base 32, using
- base change [i] based on digital (28, 85, 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]

(45, 89, 740)-Net over F₃₂ — Digital

Digital (45, 89, 740)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32⁸⁹, 740, F₃₂, 44) (dual of [740, 651, 45]-code), using
- discarding factors / shortening the dual code based on linear OA(32⁸⁹, 1041, F₃₂, 44) (dual of [1041, 952, 45]-code), using
  - constructionÂ X applied to C_e(43) âŠ‚ C_e(37) [i] based on
    1. linear OA(32⁸⁴, 1024, F₃₂, 44) (dual of [1024, 940, 45]-code), using an extension C_e(43) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [1,43], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 44 [i]
    2. 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]
    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]

(45, 89, 358503)-Net in Base 32 — Upper bound on s

There is no (45, 89, 358504)-net in base 32, because

the generalized Rao bound for nets shows that 32^mÂ â‰¥ 90 856821 276530 606339 012661 407569 709047 366063 350318 932508 736186 842254 651256 796856 445515 359102 695393 899856 486714 282227 453834 192460 790214 > 32⁸⁹ [i]