MinT - Best Known (105−50, 105, s)-Nets in Base 32

Best Known (105−50, 105, s)-Nets in Base 32

(105−50, 105, 240)-Net over F₃₂ — Constructive and digital

Digital (55, 105, 240)-net over F₃₂, using

t-expansion [i] based on digital (51, 105, 240)-net over F₃₂, using
- 4 times m-reduction [i] based on digital (51, 109, 240)-net over F₃₂, using
  - (u,Â u+v)-construction [i] based on
    1. digital (11, 40, 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, 69, 120)-net over F₃₂, using
      - net from sequence [i] based on digital (11, 119)-sequence over F₃₂ (see above)

(105−50, 105, 513)-Net in Base 32 — Constructive

(55, 105, 513)-net in base 32, using

t-expansion [i] based on (46, 105, 513)-net in base 32, using
- 3 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]

(105−50, 105, 1081)-Net over F₃₂ — Digital

Digital (55, 105, 1081)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32¹⁰⁵, 1081, F₃₂, 50) (dual of [1081, 976, 51]-code), using
- discarding factors / shortening the dual code based on linear OA(32¹⁰⁵, 1096, F₃₂, 50) (dual of [1096, 991, 51]-code), using
  - 60 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (5, 0, 1, 0, 0, 1, 7 times 0, 1, 16 times 0, 1, 29 times 0) [i] based on linear OA(32⁹⁶, 1027, F₃₂, 50) (dual of [1027, 931, 51]-code), using
    - constructionÂ XX applied to C₁Â =Â C([1022,47]), C₂Â =Â C([0,48]), C₃Â =Â C₁Â +Â C₂Â =Â C([0,47]), and C_âˆ©Â =Â C₁Â âˆ©Â C₂Â =Â C([1022,48]) [i] based on
      1. linear OA(32⁹⁴, 1023, F₃₂, 49) (dual of [1023, 929, 50]-code), using the primitive BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â {−1,0,…,47}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 50 [i]
      2. linear OA(32⁹⁴, 1023, F₃₂, 49) (dual of [1023, 929, 50]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [0,48], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 50 [i]
      3. linear OA(32⁹⁶, 1023, F₃₂, 50) (dual of [1023, 927, 51]-code), using the primitive BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â {−1,0,…,48}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 51 [i]
      4. linear OA(32⁹², 1023, F₃₂, 48) (dual of [1023, 931, 49]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [0,47], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 49 [i]
      5. linear OA(32⁰, 2, F₃₂, 0) (dual of [2, 2, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(32⁰, s, F₃₂, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]
      6. linear OA(32⁰, 2, F₃₂, 0) (dual of [2, 2, 1]-code) (see above)

(105−50, 105, 688471)-Net in Base 32 — Upper bound on s

There is no (55, 105, 688472)-net in base 32, because

the generalized Rao bound for nets shows that 32^mÂ â‰¥ 109 838775 421849 973335 513825 533702 158657 920880 797168 090786 939205 117344 097472 001831 698035 621467 098779 187396 762966 777479 113670 603175 110865 688558 201100 801695 574040 > 32¹⁰⁵ [i]

Best Known (105−50, 105, s)-Nets in Base 32

(105−50, 105, 240)-Net over F32 — Constructive and digital

(105−50, 105, 513)-Net in Base 32 — Constructive

(105−50, 105, 1081)-Net over F32 — Digital

(105−50, 105, 688471)-Net in Base 32 — Upper bound on s

(105−50, 105, 240)-Net over F₃₂ — Constructive and digital

(105−50, 105, 1081)-Net over F₃₂ — Digital