MinT - Best Known (68, 68+23, s)-Nets in Base 5

Best Known (68, 68+23, s)-Nets in Base 5

(68, 68+23, 306)-Net over F₅ — Constructive and digital

Digital (68, 91, 306)-net over F₅, using

5¹ times duplication [i] based on digital (67, 90, 306)-net over F₅, using
- (u,Â u+v)-construction [i] based on
  1. digital (13, 24, 54)-net over F₅, using
    - trace code for nets [i] based on digital (1, 12, 27)-net over F₂₅, using
      - net from sequence [i] based on digital (1, 26)-sequence over F₂₅, using
        Niederreiter sequence [i]
  2. digital (43, 66, 252)-net over F₅, using
    - trace code for nets [i] based on digital (10, 33, 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]

(68, 68+23, 2133)-Net over F₅ — Digital

Digital (68, 91, 2133)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5⁹¹, 2133, F₅, 23) (dual of [2133, 2042, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(5⁹¹, 3125, F₅, 23) (dual of [3125, 3034, 24]-code), using
  - an extension C_e(22) of the primitive narrow-sense BCH-code C(I) with length 3124Â = 5⁵âˆ’1, defining interval IÂ =Â [1,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]

(68, 68+23, 642383)-Net in Base 5 — Upper bound on s

There is no (68, 91, 642384)-net in base 5, because

1 times m-reduction [i] would yield (68, 90, 642384)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 807 799703 386079 017572 699905 076598 974675 554904 377604 300942 268225 > 5⁹⁰ [i]