MinT - Best Known (82−25, 82, s)-Nets in Base 5

Best Known (82−25, 82, s)-Nets in Base 5

(82−25, 82, 258)-Net over F₅ — Constructive and digital

Digital (57, 82, 258)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (0, 12, 6)-net over F₅, using
  - net from sequence [i] based on digital (0, 5)-sequence over F₅, using
    - Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 0 and N(F) ≥ 6, using
      - the rational function field F₅(x) [i]
    - Niederreiter sequence [i]
2. digital (45, 70, 252)-net over F₅, using
  - trace code for nets [i] based on digital (10, 35, 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]

(82−25, 82, 640)-Net over F₅ — Digital

Digital (57, 82, 640)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5⁸², 640, F₅, 25) (dual of [640, 558, 26]-code), using
- 7 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (1, 6 times 0) [i] based on linear OA(5⁸¹, 632, F₅, 25) (dual of [632, 551, 26]-code), using
  - constructionÂ XX applied to C₁Â =Â C([623,22]), C₂Â =Â C([0,23]), C₃Â =Â C₁Â +Â C₂Â =Â C([0,22]), and C_âˆ©Â =Â C₁Â âˆ©Â C₂Â =Â C([623,23]) [i] based on
    1. linear OA(5⁷⁷, 624, F₅, 24) (dual of [624, 547, 25]-code), using the primitive BCH-code C(I) with length 624Â = 5⁴âˆ’1, defining interval IÂ =Â {−1,0,…,22}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 25 [i]
    2. linear OA(5⁷⁷, 624, F₅, 24) (dual of [624, 547, 25]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624Â = 5⁴âˆ’1, defining interval IÂ =Â [0,23], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 25 [i]
    3. linear OA(5⁸¹, 624, F₅, 25) (dual of [624, 543, 26]-code), using the primitive BCH-code C(I) with length 624Â = 5⁴âˆ’1, defining interval IÂ =Â {−1,0,…,23}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 26 [i]
    4. linear OA(5⁷³, 624, F₅, 23) (dual of [624, 551, 24]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624Â = 5⁴âˆ’1, defining interval IÂ =Â [0,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 24 [i]
    5. linear OA(5⁰, 4, F₅, 0) (dual of [4, 4, 1]-code), using
      - discarding factors / shortening the dual code based on linear OA(5⁰, s, F₅, 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(5⁰, 4, F₅, 0) (dual of [4, 4, 1]-code) (see above)

(82−25, 82, 69070)-Net in Base 5 — Upper bound on s

There is no (57, 82, 69071)-net in base 5, because

1 times m-reduction [i] would yield (57, 81, 69071)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 413 592951 097177 909867 588562 123286 018257 062971 446272 097745 > 5⁸¹ [i]

Best Known (82−25, 82, s)-Nets in Base 5

(82−25, 82, 258)-Net over F5 — Constructive and digital

(82−25, 82, 640)-Net over F5 — Digital

(82−25, 82, 69070)-Net in Base 5 — Upper bound on s

(82−25, 82, 258)-Net over F₅ — Constructive and digital

(82−25, 82, 640)-Net over F₅ — Digital