MinT - Best Known (30−12, 30, s)-Nets in Base 32

Best Known (30−12, 30, s)-Nets in Base 32

(30−12, 30, 231)-Net over F₃₂ — Constructive and digital

Digital (18, 30, 231)-net over F₃₂, using

1 times m-reduction [i] based on digital (18, 31, 231)-net over F₃₂, using
- generalized (u,Â u+v)-construction [i] based on
  1. digital (0, 1, 33)-net over F₃₂, using
    - s-reduction based on digital (0, 1, s)-net over F₃₂ with arbitrarily large s, using
      - construction for strength kÂ =Â 1 [i]
  2. digital (0, 2, 33)-net over F₃₂, using
    - kÂ =Â 2 construction [i]
  3. digital (0, 2, 33)-net over F₃₂ (see above)
  4. digital (0, 3, 33)-net over F₃₂, using
    - net from sequence [i] based on digital (0, 32)-sequence over F₃₂, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 0 and N(F) ≥ 33, using
        the rational function field F₃₂(x) [i]
      - Niederreiter sequence [i]
  5. digital (0, 4, 33)-net over F₃₂, using
    - net from sequence [i] based on digital (0, 32)-sequence over F₃₂ (see above)
  6. digital (0, 6, 33)-net over F₃₂, using
    - net from sequence [i] based on digital (0, 32)-sequence over F₃₂ (see above)
  7. digital (0, 13, 33)-net over F₃₂, using
    - net from sequence [i] based on digital (0, 32)-sequence over F₃₂ (see above)

(30−12, 30, 684)-Net in Base 32 — Constructive

(18, 30, 684)-net in base 32, using

base change [i] based on digital (13, 25, 684)-net over F₆₄, using
- net defined by OOA [i] based on linear OOA(64²⁵, 684, F₆₄, 12, 12) (dual of [(684, 12), 8183, 13]-NRT-code), using
  - OA 6-folding and stacking [i] based on linear OA(64²⁵, 4104, F₆₄, 12) (dual of [4104, 4079, 13]-code), using
    - constructionÂ X applied to C_e(11) âŠ‚ C_e(8) [i] based on
      1. linear OA(64²³, 4096, F₆₄, 12) (dual of [4096, 4073, 13]-code), using an extension C_e(11) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]
      2. linear OA(64¹⁷, 4096, F₆₄, 9) (dual of [4096, 4079, 10]-code), using an extension C_e(8) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,8], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]
      3. linear OA(64², 8, F₆₄, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,64)), using
        discarding factors / shortening the dual code based on linear OA(64², 64, F₆₄, 2) (dual of [64, 62, 3]-code or 64-arc in PG(1,64)), using
        Reedâ€“Solomon code RS(62,64) [i]

(30−12, 30, 2026)-Net over F₃₂ — Digital

Digital (18, 30, 2026)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32³⁰, 2026, F₃₂, 12) (dual of [2026, 1996, 13]-code), using
- 992 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (3, 5 times 0, 1, 31 times 0, 1, 125 times 0, 1, 307 times 0, 1, 519 times 0) [i] based on linear OA(32²³, 1027, F₃₂, 12) (dual of [1027, 1004, 13]-code), using
  - constructionÂ XX applied to C₁Â =Â C([1022,9]), C₂Â =Â C([0,10]), C₃Â =Â C₁Â +Â C₂Â =Â C([0,9]), and C_âˆ©Â =Â C₁Â âˆ©Â C₂Â =Â C([1022,10]) [i] based on
    1. linear OA(32²¹, 1023, F₃₂, 11) (dual of [1023, 1002, 12]-code), using the primitive BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â {−1,0,…,9}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]
    2. linear OA(32²¹, 1023, F₃₂, 11) (dual of [1023, 1002, 12]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [0,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]
    3. linear OA(32²³, 1023, F₃₂, 12) (dual of [1023, 1000, 13]-code), using the primitive BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â {−1,0,…,10}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]
    4. linear OA(32¹⁹, 1023, F₃₂, 10) (dual of [1023, 1004, 11]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [0,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [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)

(30−12, 30, 2052)-Net in Base 32

(18, 30, 2052)-net in base 32, using

base change [i] based on digital (13, 25, 2052)-net over F₆₄, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(64²⁵, 2052, F₆₄, 2, 12) (dual of [(2052, 2), 4079, 13]-NRT-code), using
  - OOA 2-folding [i] based on linear OA(64²⁵, 4104, F₆₄, 12) (dual of [4104, 4079, 13]-code), using
    - constructionÂ X applied to C_e(11) âŠ‚ C_e(8) [i] based on
      1. linear OA(64²³, 4096, F₆₄, 12) (dual of [4096, 4073, 13]-code), using an extension C_e(11) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]
      2. linear OA(64¹⁷, 4096, F₆₄, 9) (dual of [4096, 4079, 10]-code), using an extension C_e(8) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,8], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]
      3. linear OA(64², 8, F₆₄, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,64)), using
        discarding factors / shortening the dual code based on linear OA(64², 64, F₆₄, 2) (dual of [64, 62, 3]-code or 64-arc in PG(1,64)), using
        Reedâ€“Solomon code RS(62,64) [i]

(30−12, 30, 3240484)-Net in Base 32 — Upper bound on s

There is no (18, 30, 3240485)-net in base 32, because

the generalized Rao bound for nets shows that 32^mÂ â‰¥ 1427 249584 888294 794553 828694 716444 850621 442448 > 32³⁰ [i]