MinT - Best Known (82−37, 82, s)-Nets in Base 27

Best Known (82−37, 82, s)-Nets in Base 27

(82−37, 82, 210)-Net over F₂₇ — Constructive and digital

Digital (45, 82, 210)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (4, 16, 64)-net over F₂₇, using
  - net from sequence [i] based on digital (4, 63)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 4 and N(F) ≥ 64, using
      - a function field by SÃ©mirat [i]
2. digital (4, 22, 64)-net over F₂₇, using
  - net from sequence [i] based on digital (4, 63)-sequence over F₂₇ (see above)
3. digital (7, 44, 82)-net over F₂₇, using
  - net from sequence [i] based on digital (7, 81)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 7 and N(F) ≥ 82, using
      - a function field by SÃ©mirat [i]

(82−37, 82, 370)-Net in Base 27 — Constructive

(45, 82, 370)-net in base 27, using

t-expansion [i] based on (43, 82, 370)-net in base 27, using
- 26 times m-reduction [i] based on (43, 108, 370)-net in base 27, using
  - base change [i] based on digital (16, 81, 370)-net over F₈₁, using
    - net from sequence [i] based on digital (16, 369)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 16 and N(F) ≥ 370, using
        a function field by GarcÃa and Quoos [i]

(82−37, 82, 1019)-Net over F₂₇ — Digital

Digital (45, 82, 1019)-net over F₂₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(27⁸², 1019, F₂₇, 37) (dual of [1019, 937, 38]-code), using
- 275 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (5, 0, 1, 0, 0, 0, 1, 9 times 0, 1, 17 times 0, 1, 32 times 0, 1, 51 times 0, 1, 70 times 0, 1, 84 times 0) [i] based on linear OA(27⁷⁰, 732, F₂₇, 37) (dual of [732, 662, 38]-code), using
  - constructionÂ XX applied to C₁Â =Â C([727,34]), 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([727,35]) [i] based on
    1. linear OA(27⁶⁸, 728, F₂₇, 36) (dual of [728, 660, 37]-code), using the primitive BCH-code C(I) with length 728Â = 27²âˆ’1, defining interval IÂ =Â {−1,0,…,34}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 37 [i]
    2. linear OA(27⁶⁸, 728, F₂₇, 36) (dual of [728, 660, 37]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728Â = 27²âˆ’1, defining interval IÂ =Â [0,35], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 37 [i]
    3. linear OA(27⁷⁰, 728, F₂₇, 37) (dual of [728, 658, 38]-code), using the primitive BCH-code C(I) with length 728Â = 27²âˆ’1, defining interval IÂ =Â {−1,0,…,35}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 38 [i]
    4. linear OA(27⁶⁶, 728, F₂₇, 35) (dual of [728, 662, 36]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728Â = 27²âˆ’1, defining interval IÂ =Â [0,34], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 36 [i]
    5. linear OA(27⁰, 2, F₂₇, 0) (dual of [2, 2, 1]-code), using
      - discarding factors / shortening the dual code based on linear OA(27⁰, s, F₂₇, 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(27⁰, 2, F₂₇, 0) (dual of [2, 2, 1]-code) (see above)

(82−37, 82, 802211)-Net in Base 27 — Upper bound on s

There is no (45, 82, 802212)-net in base 27, because

1 times m-reduction [i] would yield (45, 81, 802212)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 87 191388 401522 381219 081879 955768 342890 247938 212267 982217 267245 416434 774508 385564 675047 061545 437806 369617 751701 119225 > 27⁸¹ [i]

Best Known (82−37, 82, s)-Nets in Base 27

(82−37, 82, 210)-Net over F27 — Constructive and digital

(82−37, 82, 370)-Net in Base 27 — Constructive

(82−37, 82, 1019)-Net over F27 — Digital

(82−37, 82, 802211)-Net in Base 27 — Upper bound on s

(82−37, 82, 210)-Net over F₂₇ — Constructive and digital

(82−37, 82, 1019)-Net over F₂₇ — Digital