MinT - Best Known (64, 64+22, s)-Nets in Base 5

Best Known (64, 64+22, s)-Nets in Base 5

(64, 64+22, 304)-Net over F₅ — Constructive and digital

Digital (64, 86, 304)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (11, 22, 52)-net over F₅, using
  - trace code for nets [i] based on digital (0, 11, 26)-net over F₂₅, using
    - net from sequence [i] based on digital (0, 25)-sequence over F₂₅, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 0 and N(F) ≥ 26, using
        the rational function field F₂₅(x) [i]
      - Niederreiter sequence [i]
2. digital (42, 64, 252)-net over F₅, using
  - trace code for nets [i] based on digital (10, 32, 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]

(64, 64+22, 1927)-Net over F₅ — Digital

Digital (64, 86, 1927)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5⁸⁶, 1927, F₅, 22) (dual of [1927, 1841, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(5⁸⁶, 3125, F₅, 22) (dual of [3125, 3039, 23]-code), using
  - an extension C_e(21) of the primitive narrow-sense BCH-code C(I) with length 3124Â = 5⁵âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]

(64, 64+22, 357782)-Net in Base 5 — Upper bound on s

There is no (64, 86, 357783)-net in base 5, because

the generalized Rao bound for nets shows that 5^mÂ â‰¥ 1 292481 630728 603440 699171 657800 354909 940934 197503 986414 251445 > 5⁸⁶ [i]