MinT - Best Known (83−40, 83, s)-Nets in Base 32

Best Known (83−40, 83, s)-Nets in Base 32

(83−40, 83, 240)-Net over F₃₂ — Constructive and digital

Digital (43, 83, 240)-net over F₃₂, using

2 times m-reduction [i] based on digital (43, 85, 240)-net over F₃₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (11, 32, 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, 53, 120)-net over F₃₂, using
    - net from sequence [i] based on digital (11, 119)-sequence over F₃₂ (see above)

(83−40, 83, 513)-Net in Base 32 — Constructive

(43, 83, 513)-net in base 32, using

7 times m-reduction [i] based on (43, 90, 513)-net in base 32, using
- base change [i] based on digital (28, 75, 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]

(83−40, 83, 840)-Net over F₃₂ — Digital

Digital (43, 83, 840)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32⁸³, 840, F₃₂, 40) (dual of [840, 757, 41]-code), using
- discarding factors / shortening the dual code based on linear OA(32⁸³, 1045, F₃₂, 40) (dual of [1045, 962, 41]-code), using
  - constructionÂ XX applied to C₁Â =Â C([1018,33]), C₂Â =Â C([3,34]), C₃Â =Â C₁Â +Â C₂Â =Â C([3,33]), and C_âˆ©Â =Â C₁Â âˆ©Â C₂Â =Â C([1018,34]) [i] based on
    1. linear OA(32⁷⁴, 1023, F₃₂, 39) (dual of [1023, 949, 40]-code), using the primitive BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â {−5,−4,…,33}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 40 [i]
    2. linear OA(32⁶³, 1023, F₃₂, 32) (dual of [1023, 960, 33]-code), using the primitive BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â {3,4,…,34}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 33 [i]
    3. linear OA(32⁷⁶, 1023, F₃₂, 40) (dual of [1023, 947, 41]-code), using the primitive BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â {−5,−4,…,34}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 41 [i]
    4. linear OA(32⁶¹, 1023, F₃₂, 31) (dual of [1023, 962, 32]-code), using the primitive BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â {3,4,…,33}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 32 [i]
    5. linear OA(32⁷, 20, F₃₂, 7) (dual of [20, 13, 8]-code or 20-arc in PG(6,32)), using
      - discarding factors / shortening the dual code based on linear OA(32⁷, 32, F₃₂, 7) (dual of [32, 25, 8]-code or 32-arc in PG(6,32)), using
        Reedâ€“Solomon code RS(25,32) [i]
    6. 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]

(83−40, 83, 472397)-Net in Base 32 — Upper bound on s

There is no (43, 83, 472398)-net in base 32, because

the generalized Rao bound for nets shows that 32^mÂ â‰¥ 84616 839348 128450 133930 050114 808428 265805 646902 966966 128065 069051 161266 524643 106000 756306 389579 967384 186813 392717 012973 695196 > 32⁸³ [i]

Best Known (83−40, 83, s)-Nets in Base 32

(83−40, 83, 240)-Net over F32 — Constructive and digital

(83−40, 83, 513)-Net in Base 32 — Constructive

(83−40, 83, 840)-Net over F32 — Digital

(83−40, 83, 472397)-Net in Base 32 — Upper bound on s

(83−40, 83, 240)-Net over F₃₂ — Constructive and digital

(83−40, 83, 840)-Net over F₃₂ — Digital