MinT - Best Known (40, 73, s)-Nets in Base 32

Best Known (40, 73, s)-Nets in Base 32

(40, 73, 240)-Net over F₃₂ — Constructive and digital

Digital (40, 73, 240)-net over F₃₂, using

3 times m-reduction [i] based on digital (40, 76, 240)-net over F₃₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (11, 29, 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, 47, 120)-net over F₃₂, using
    - net from sequence [i] based on digital (11, 119)-sequence over F₃₂ (see above)

(40, 73, 327)-Net in Base 32 — Constructive

(40, 73, 327)-net in base 32, using

32¹ times duplication [i] based on (39, 72, 327)-net in base 32, using
- (u,Â u+v)-construction [i] based on
  1. (8, 24, 150)-net in base 32, using
    - base change [i] based on (4, 20, 150)-net in base 64, using
      - 1 times m-reduction [i] based on (4, 21, 150)-net in base 64, using
        base change [i] based on digital (1, 18, 150)-net over F₁₂₈, using
        net from sequence [i] based on digital (1, 149)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 1 and N(F) ≥ 150, using
        a function field by SÃ©mirat [i]
  2. (15, 48, 177)-net in base 32, using
    - base change [i] based on digital (7, 40, 177)-net over F₆₄, using
      - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
        a function field by GarcÃa and Quoos [i]

(40, 73, 1171)-Net over F₃₂ — Digital

Digital (40, 73, 1171)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32⁷³, 1171, F₃₂, 33) (dual of [1171, 1098, 34]-code), using
- 137 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (5, 0, 1, 0, 0, 0, 1, 8 times 0, 1, 19 times 0, 1, 35 times 0, 1, 65 times 0) [i] based on linear OA(32⁶³, 1024, F₃₂, 33) (dual of [1024, 961, 34]-code), using
  - an extension C_e(32) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [1,32], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 33 [i]

(40, 73, 1301215)-Net in Base 32 — Upper bound on s

There is no (40, 73, 1301216)-net in base 32, because

1 times m-reduction [i] would yield (40, 72, 1301216)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 2 348554 440763 228857 090715 869755 974853 866845 030367 535749 771683 261346 833880 128425 933140 190413 503747 066079 480671 > 32⁷² [i]