MinT - Best Known (89−43, 89, s)-Nets in Base 32

Best Known (89−43, 89, s)-Nets in Base 32

(89−43, 89, 240)-Net over F₃₂ — Constructive and digital

Digital (46, 89, 240)-net over F₃₂, using

5 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)

(89−43, 89, 513)-Net in Base 32 — Constructive

(46, 89, 513)-net in base 32, using

19 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]

(89−43, 89, 866)-Net over F₃₂ — Digital

Digital (46, 89, 866)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32⁸⁹, 866, F₃₂, 43) (dual of [866, 777, 44]-code), using
- discarding factors / shortening the dual code based on linear OA(32⁸⁹, 1047, F₃₂, 43) (dual of [1047, 958, 44]-code), using
  - constructionÂ X applied to C_e(42) âŠ‚ C_e(34) [i] based on
    1. linear OA(32⁸², 1024, F₃₂, 43) (dual of [1024, 942, 44]-code), using an extension C_e(42) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [1,42], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 43 [i]
    2. linear OA(32⁶⁶, 1024, F₃₂, 35) (dual of [1024, 958, 36]-code), using an extension C_e(34) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [1,34], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 35 [i]
    3. linear OA(32⁷, 23, F₃₂, 7) (dual of [23, 16, 8]-code or 23-arc in PG(6,32)), using
      - discarding factors / shortening the dual code based on linear OA(32⁷, 32, F₃₂, 7) (dual of [32, 25, 8]-code or 32-arc in PG(6,32)), using
        Reedâ€“Solomon code RS(25,32) [i]

(89−43, 89, 568090)-Net in Base 32 — Upper bound on s

There is no (46, 89, 568091)-net in base 32, because

1 times m-reduction [i] would yield (46, 88, 568091)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 2 839220 760889 431087 371836 947399 003780 646354 278821 915392 330350 742908 083495 655313 886820 192256 302887 119123 209102 618275 810923 192816 204684 > 32⁸⁸ [i]