MinT - Best Known (91−17, 91, s)-Nets in Base 5

Best Known (91−17, 91, s)-Nets in Base 5

(91−17, 91, 1971)-Net over F₅ — Constructive and digital

Digital (74, 91, 1971)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (4, 12, 18)-net over F₅, using
  - net from sequence [i] based on digital (4, 17)-sequence over F₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 4 and N(F) ≥ 18, using
      - an explicitly constructive algebraic function field [i]
2. digital (62, 79, 1953)-net over F₅, using
  - net defined by OOA [i] based on linear OOA(5⁷⁹, 1953, F₅, 17, 17) (dual of [(1953, 17), 33122, 18]-NRT-code), using
    - OOA 8-folding and stacking with additional row [i] based on linear OA(5⁷⁹, 15625, F₅, 17) (dual of [15625, 15546, 18]-code), using
      - an extension C_e(16) of the primitive narrow-sense BCH-code C(I) with length 15624Â = 5⁶âˆ’1, defining interval IÂ =Â [1,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]

(91−17, 91, 16072)-Net over F₅ — Digital

Digital (74, 91, 16072)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(91−17, 91, large)-Net in Base 5 — Upper bound on s

There is no (74, 91, large)-net in base 5, because

15 times m-reduction [i] would yield (74, 76, large)-net in base 5, but
- mutually orthogonal hypercube bound [i]