MinT - Best Known (89−41, 89, s)-Nets in Base 32

Best Known (89−41, 89, s)-Nets in Base 32

(89−41, 89, 240)-Net over F₃₂ — Constructive and digital

Digital (48, 89, 240)-net over F₃₂, using

11 times m-reduction [i] based on digital (48, 100, 240)-net over F₃₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (11, 37, 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, 63, 120)-net over F₃₂, using
    - net from sequence [i] based on digital (11, 119)-sequence over F₃₂ (see above)

(89−41, 89, 513)-Net in Base 32 — Constructive

(48, 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]

(89−41, 89, 1173)-Net over F₃₂ — Digital

Digital (48, 89, 1173)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32⁸⁹, 1173, F₃₂, 41) (dual of [1173, 1084, 42]-code), using
- 133 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (5, 0, 1, 0, 0, 0, 1, 9 times 0, 1, 18 times 0, 1, 35 times 0, 1, 61 times 0) [i] based on linear OA(32⁷⁹, 1030, F₃₂, 41) (dual of [1030, 951, 42]-code), using
  - constructionÂ XX applied to C₁Â =Â C([1021,37]), C₂Â =Â C([0,38]), C₃Â =Â C₁Â +Â C₂Â =Â C([0,37]), and C_âˆ©Â =Â C₁Â âˆ©Â C₂Â =Â C([1021,38]) [i] based on
    1. 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 {−2,−1,…,37}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 41 [i]
    2. linear OA(32⁷⁴, 1023, F₃₂, 39) (dual of [1023, 949, 40]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [0,38], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 40 [i]
    3. 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 {−2,−1,…,38}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 42 [i]
    4. linear OA(32⁷², 1023, F₃₂, 38) (dual of [1023, 951, 39]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [0,37], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 39 [i]
    5. linear OA(32¹, 5, F₃₂, 1) (dual of [5, 4, 2]-code), using
      - discarding factors / shortening the dual code based on linear OA(32¹, 32, F₃₂, 1) (dual of [32, 31, 2]-code), using
        Reedâ€“Solomon code RS(31,32) [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]

(89−41, 89, 1123570)-Net in Base 32 — Upper bound on s

There is no (48, 89, 1123571)-net in base 32, because

1 times m-reduction [i] would yield (48, 88, 1123571)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 2 839213 905480 619280 533299 207893 422503 327113 941040 097662 251905 643299 676443 909757 956640 546434 796676 057440 025057 513740 853816 148797 727656 > 32⁸⁸ [i]

Best Known (89−41, 89, s)-Nets in Base 32

(89−41, 89, 240)-Net over F32 — Constructive and digital

(89−41, 89, 513)-Net in Base 32 — Constructive

(89−41, 89, 1173)-Net over F32 — Digital

(89−41, 89, 1123570)-Net in Base 32 — Upper bound on s

(89−41, 89, 240)-Net over F₃₂ — Constructive and digital

(89−41, 89, 1173)-Net over F₃₂ — Digital