MinT - Best Known (89−42, 89, s)-Nets in Base 32

Best Known (89−42, 89, s)-Nets in Base 32

(89−42, 89, 240)-Net over F₃₂ — Constructive and digital

Digital (47, 89, 240)-net over F₃₂, using

8 times m-reduction [i] based on digital (47, 97, 240)-net over F₃₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (11, 36, 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, 61, 120)-net over F₃₂, using
    - net from sequence [i] based on digital (11, 119)-sequence over F₃₂ (see above)

(89−42, 89, 513)-Net in Base 32 — Constructive

(47, 89, 513)-net in base 32, using

t-expansion [i] based on (46, 89, 513)-net in base 32, using
- 19 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]

(89−42, 89, 1023)-Net over F₃₂ — Digital

Digital (47, 89, 1023)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32⁸⁹, 1023, F₃₂, 42) (dual of [1023, 934, 43]-code), using
- discarding factors / shortening the dual code based on linear OA(32⁸⁹, 1051, F₃₂, 42) (dual of [1051, 962, 43]-code), using
  - constructionÂ XX applied to C₁Â =Â C([1016,33]), 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([1016,34]) [i] based on
    1. linear OA(32⁷⁸, 1023, F₃₂, 41) (dual of [1023, 945, 42]-code), using the primitive BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â {−7,−6,…,33}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 42 [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(32⁸⁰, 1023, F₃₂, 42) (dual of [1023, 943, 43]-code), using the primitive BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â {−7,−6,…,34}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 43 [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(32⁹, 26, F₃₂, 9) (dual of [26, 17, 10]-code or 26-arc in PG(8,32)), using
      - discarding factors / shortening the dual code based on linear OA(32⁹, 32, F₃₂, 9) (dual of [32, 23, 10]-code or 32-arc in PG(8,32)), using
        Reedâ€“Solomon code RS(23,32) [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]

(89−42, 89, 670027)-Net in Base 32 — Upper bound on s

There is no (47, 89, 670028)-net in base 32, because

the generalized Rao bound for nets shows that 32^mÂ â‰¥ 90 855031 913291 640173 234780 920312 012819 379656 869035 501807 142054 121184 472023 286515 822932 659833 339287 364103 304036 865455 897287 881441 196504 > 32⁸⁹ [i]

Best Known (89−42, 89, s)-Nets in Base 32

(89−42, 89, 240)-Net over F32 — Constructive and digital

(89−42, 89, 513)-Net in Base 32 — Constructive

(89−42, 89, 1023)-Net over F32 — Digital

(89−42, 89, 670027)-Net in Base 32 — Upper bound on s

(89−42, 89, 240)-Net over F₃₂ — Constructive and digital

(89−42, 89, 1023)-Net over F₃₂ — Digital