MinT - Best Known (82−37, 82, s)-Nets in Base 32

Best Known (82−37, 82, s)-Nets in Base 32

(82−37, 82, 240)-Net over F₃₂ — Constructive and digital

Digital (45, 82, 240)-net over F₃₂, using

9 times m-reduction [i] based on digital (45, 91, 240)-net over F₃₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (11, 34, 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, 57, 120)-net over F₃₂, using
    - net from sequence [i] based on digital (11, 119)-sequence over F₃₂ (see above)

(82−37, 82, 513)-Net in Base 32 — Constructive

(45, 82, 513)-net in base 32, using

20 times m-reduction [i] based on (45, 102, 513)-net in base 32, using
- base change [i] based on digital (28, 85, 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]

(82−37, 82, 1261)-Net over F₃₂ — Digital

Digital (45, 82, 1261)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32⁸², 1261, F₃₂, 37) (dual of [1261, 1179, 38]-code), using
- 222 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, 1, 60 times 0, 1, 90 times 0) [i] based on linear OA(32⁷⁰, 1027, F₃₂, 37) (dual of [1027, 957, 38]-code), using
  - constructionÂ XX applied to C₁Â =Â C([1022,34]), C₂Â =Â C([0,35]), C₃Â =Â C₁Â +Â C₂Â =Â C([0,34]), and C_âˆ©Â =Â C₁Â âˆ©Â C₂Â =Â C([1022,35]) [i] based on
    1. linear OA(32⁶⁸, 1023, F₃₂, 36) (dual of [1023, 955, 37]-code), using the primitive BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â {−1,0,…,34}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 37 [i]
    2. 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]
    3. 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]
    4. linear OA(32⁶⁶, 1023, F₃₂, 35) (dual of [1023, 957, 36]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [0,34], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 36 [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)

(82−37, 82, 1445241)-Net in Base 32 — Upper bound on s

There is no (45, 82, 1445242)-net in base 32, because

1 times m-reduction [i] would yield (45, 81, 1445242)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 82 632124 488929 986462 584475 476340 686645 957814 713760 423685 499329 824765 444409 227215 172210 226468 064973 389071 502877 687088 954660 > 32⁸¹ [i]

Best Known (82−37, 82, s)-Nets in Base 32

(82−37, 82, 240)-Net over F32 — Constructive and digital

(82−37, 82, 513)-Net in Base 32 — Constructive

(82−37, 82, 1261)-Net over F32 — Digital

(82−37, 82, 1445241)-Net in Base 32 — Upper bound on s

(82−37, 82, 240)-Net over F₃₂ — Constructive and digital

(82−37, 82, 1261)-Net over F₃₂ — Digital