MinT - Best Known (82−38, 82, s)-Nets in Base 32

Best Known (82−38, 82, s)-Nets in Base 32

(82−38, 82, 240)-Net over F₃₂ — Constructive and digital

Digital (44, 82, 240)-net over F₃₂, using

6 times m-reduction [i] based on digital (44, 88, 240)-net over F₃₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (11, 33, 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, 55, 120)-net over F₃₂, using
    - net from sequence [i] based on digital (11, 119)-sequence over F₃₂ (see above)

(82−38, 82, 513)-Net in Base 32 — Constructive

(44, 82, 513)-net in base 32, using

14 times m-reduction [i] based on (44, 96, 513)-net in base 32, using
- base change [i] based on digital (28, 80, 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]

(82−38, 82, 1105)-Net over F₃₂ — Digital

Digital (44, 82, 1105)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32⁸², 1105, F₃₂, 38) (dual of [1105, 1023, 39]-code), using
- discarding factors / shortening the dual code based on linear OA(32⁸², 1107, F₃₂, 38) (dual of [1107, 1025, 39]-code), using
  - 70 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (6, 0, 1, 0, 0, 0, 1, 9 times 0, 1, 17 times 0, 1, 35 times 0) [i] based on linear OA(32⁷², 1027, F₃₂, 38) (dual of [1027, 955, 39]-code), using
    - constructionÂ XX applied to C₁Â =Â C([1022,35]), C₂Â =Â C([0,36]), C₃Â =Â C₁Â +Â C₂Â =Â C([0,35]), and C_âˆ©Â =Â C₁Â âˆ©Â C₂Â =Â C([1022,36]) [i] based on
      1. linear OA(32⁷⁰, 1023, F₃₂, 37) (dual of [1023, 953, 38]-code), using the primitive BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â {−1,0,…,35}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 38 [i]
      2. linear OA(32⁷⁰, 1023, F₃₂, 37) (dual of [1023, 953, 38]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [0,36], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 38 [i]
      3. linear OA(32⁷², 1023, F₃₂, 38) (dual of [1023, 951, 39]-code), using the primitive BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â {−1,0,…,36}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 39 [i]
      4. linear OA(32⁶⁸, 1023, F₃₂, 36) (dual of [1023, 955, 37]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [0,35], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 37 [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)

(82−38, 82, 801233)-Net in Base 32 — Upper bound on s

There is no (44, 82, 801234)-net in base 32, because

the generalized Rao bound for nets shows that 32^mÂ â‰¥ 2644 262659 469358 444290 279916 533432 231998 376786 237612 588731 684487 347401 662724 421299 378275 017344 392749 656101 139579 550472 039200 > 32⁸² [i]

Best Known (82−38, 82, s)-Nets in Base 32

(82−38, 82, 240)-Net over F32 — Constructive and digital

(82−38, 82, 513)-Net in Base 32 — Constructive

(82−38, 82, 1105)-Net over F32 — Digital

(82−38, 82, 801233)-Net in Base 32 — Upper bound on s

(82−38, 82, 240)-Net over F₃₂ — Constructive and digital

(82−38, 82, 1105)-Net over F₃₂ — Digital