MinT - Best Known (37−16, 37, s)-Nets in Base 25

Best Known (37−16, 37, s)-Nets in Base 25

(37−16, 37, 178)-Net over F₂₅ — Constructive and digital

Digital (21, 37, 178)-net over F₂₅, using

(u,Â u+v)-construction [i] based on
1. digital (3, 11, 52)-net over F₂₅, using
  - net from sequence [i] based on digital (3, 51)-sequence over F₂₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 3 and N(F) ≥ 52, using
      - a function field by GarcÃa and Quoos [i]
2. digital (10, 26, 126)-net over F₂₅, using
  - net from sequence [i] based on digital (10, 125)-sequence over F₂₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 10 and N(F) ≥ 126, using
      - the Hermitian function field over F₂₅ [i]

(37−16, 37, 784)-Net over F₂₅ — Digital

Digital (21, 37, 784)-net over F₂₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(25³⁷, 784, F₂₅, 16) (dual of [784, 747, 17]-code), using
- 150 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (3, 0, 0, 0, 1, 13 times 0, 1, 39 times 0, 1, 91 times 0) [i] based on linear OA(25³¹, 628, F₂₅, 16) (dual of [628, 597, 17]-code), using
  - constructionÂ XX applied to C₁Â =Â C([623,13]), C₂Â =Â C([0,14]), C₃Â =Â C₁Â +Â C₂Â =Â C([0,13]), and C_âˆ©Â =Â C₁Â âˆ©Â C₂Â =Â C([623,14]) [i] based on
    1. linear OA(25²⁹, 624, F₂₅, 15) (dual of [624, 595, 16]-code), using the primitive BCH-code C(I) with length 624Â = 25²âˆ’1, defining interval IÂ =Â {−1,0,…,13}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]
    2. linear OA(25²⁹, 624, F₂₅, 15) (dual of [624, 595, 16]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624Â = 25²âˆ’1, defining interval IÂ =Â [0,14], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]
    3. linear OA(25³¹, 624, F₂₅, 16) (dual of [624, 593, 17]-code), using the primitive BCH-code C(I) with length 624Â = 25²âˆ’1, defining interval IÂ =Â {−1,0,…,14}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]
    4. linear OA(25²⁷, 624, F₂₅, 14) (dual of [624, 597, 15]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624Â = 25²âˆ’1, defining interval IÂ =Â [0,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]
    5. linear OA(25⁰, 2, F₂₅, 0) (dual of [2, 2, 1]-code), using
      - discarding factors / shortening the dual code based on linear OA(25⁰, s, F₂₅, 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(25⁰, 2, F₂₅, 0) (dual of [2, 2, 1]-code) (see above)

(37−16, 37, 458086)-Net in Base 25 — Upper bound on s

There is no (21, 37, 458087)-net in base 25, because

the generalized Rao bound for nets shows that 25^mÂ â‰¥ 5293 979084 991334 165282 128313 911362 461749 967183 520065 > 25³⁷ [i]

Best Known (37−16, 37, s)-Nets in Base 25

(37−16, 37, 178)-Net over F25 — Constructive and digital

(37−16, 37, 784)-Net over F25 — Digital

(37−16, 37, 458086)-Net in Base 25 — Upper bound on s

(37−16, 37, 178)-Net over F₂₅ — Constructive and digital

(37−16, 37, 784)-Net over F₂₅ — Digital