MinT - Best Known (65, 65+19, s)-Nets in Base 5

Best Known (65, 65+19, s)-Nets in Base 5

(65, 65+19, 384)-Net over F₅ — Constructive and digital

Digital (65, 84, 384)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (17, 26, 132)-net over F₅, using
  - trace code for nets [i] based on digital (4, 13, 66)-net over F₂₅, using
    - net from sequence [i] based on digital (4, 65)-sequence over F₂₅, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 4 and N(F) ≥ 66, using
        a function field by GarcÃa and Quoos [i]
2. digital (39, 58, 252)-net over F₅, using
  - trace code for nets [i] based on digital (10, 29, 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]

(65, 65+19, 3546)-Net over F₅ — Digital

Digital (65, 84, 3546)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5⁸⁴, 3546, F₅, 19) (dual of [3546, 3462, 20]-code), using
- 408 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 0, 1, 4 times 0, 1, 12 times 0, 1, 27 times 0, 1, 57 times 0, 1, 112 times 0, 1, 188 times 0) [i] based on linear OA(5⁷⁶, 3130, F₅, 19) (dual of [3130, 3054, 20]-code), using
  - constructionÂ X applied to C_e(18) âŠ‚ C_e(17) [i] based on
    1. linear OA(5⁷⁶, 3125, F₅, 19) (dual of [3125, 3049, 20]-code), using an extension C_e(18) of the primitive narrow-sense BCH-code C(I) with length 3124Â = 5⁵âˆ’1, defining interval IÂ =Â [1,18], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]
    2. linear OA(5⁷¹, 3125, F₅, 18) (dual of [3125, 3054, 19]-code), using an extension C_e(17) of the primitive narrow-sense BCH-code C(I) with length 3124Â = 5⁵âˆ’1, defining interval IÂ =Â [1,17], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 18 [i]
    3. linear OA(5⁰, 5, F₅, 0) (dual of [5, 5, 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]

(65, 65+19, 2895657)-Net in Base 5 — Upper bound on s

There is no (65, 84, 2895658)-net in base 5, because

1 times m-reduction [i] would yield (65, 83, 2895658)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 10339 766423 896687 670424 105760 429042 148331 433221 852925 888425 > 5⁸³ [i]