MinT - Best Known (105−49, 105, s)-Nets in Base 32

Best Known (105−49, 105, s)-Nets in Base 32

(105−49, 105, 240)-Net over F₃₂ — Constructive and digital

Digital (56, 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)

(105−49, 105, 513)-Net in Base 32 — Constructive

(56, 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]

(105−49, 105, 1213)-Net over F₃₂ — Digital

Digital (56, 105, 1213)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32¹⁰⁵, 1213, F₃₂, 49) (dual of [1213, 1108, 50]-code), using
- 175 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (5, 0, 1, 0, 0, 1, 6 times 0, 1, 14 times 0, 1, 28 times 0, 1, 48 times 0, 1, 69 times 0) [i] based on linear OA(32⁹⁴, 1027, F₃₂, 49) (dual of [1027, 933, 50]-code), using
  - constructionÂ XX applied to C₁Â =Â C([1022,46]), C₂Â =Â C([0,47]), C₃Â =Â C₁Â +Â C₂Â =Â C([0,46]), and C_âˆ©Â =Â C₁Â âˆ©Â C₂Â =Â C([1022,47]) [i] based on
    1. linear OA(32⁹², 1023, F₃₂, 48) (dual of [1023, 931, 49]-code), using the primitive BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â {−1,0,…,46}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 49 [i]
    2. 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]
    3. 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]
    4. linear OA(32⁹⁰, 1023, F₃₂, 47) (dual of [1023, 933, 48]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [0,46], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 48 [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)

(105−49, 105, 1052719)-Net in Base 32 — Upper bound on s

There is no (56, 105, 1052720)-net in base 32, because

1 times m-reduction [i] would yield (56, 104, 1052720)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 3 432459 981864 879611 804340 176149 525115 835239 350441 555408 655713 875022 486885 711615 591538 540164 415396 397769 565420 235588 018006 876176 974654 818311 952146 853030 423182 > 32¹⁰⁴ [i]

Best Known (105−49, 105, s)-Nets in Base 32

(105−49, 105, 240)-Net over F32 — Constructive and digital

(105−49, 105, 513)-Net in Base 32 — Constructive

(105−49, 105, 1213)-Net over F32 — Digital

(105−49, 105, 1052719)-Net in Base 32 — Upper bound on s

(105−49, 105, 240)-Net over F₃₂ — Constructive and digital

(105−49, 105, 1213)-Net over F₃₂ — Digital