MinT - Best Known (100, 123, s)-Nets in Base 5

Best Known (100, 123, s)-Nets in Base 5

(100, 123, 1436)-Net over F₅ — Constructive and digital

Digital (100, 123, 1436)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (3, 14, 16)-net over F₅, using
  - net from sequence [i] based on digital (3, 15)-sequence over F₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 3 and N(F) ≥ 16, using
      - an explicitly constructive algebraic function field [i]
2. digital (86, 109, 1420)-net over F₅, using
  - net defined by OOA [i] based on linear OOA(5¹⁰⁹, 1420, F₅, 23, 23) (dual of [(1420, 23), 32551, 24]-NRT-code), using
    - OOA 11-folding and stacking with additional row [i] based on linear OA(5¹⁰⁹, 15621, F₅, 23) (dual of [15621, 15512, 24]-code), using
      - discarding factors / shortening the dual code based on linear OA(5¹⁰⁹, 15625, F₅, 23) (dual of [15625, 15516, 24]-code), using
        an extension C_e(22) of the primitive narrow-sense BCH-code C(I) with length 15624Â = 5⁶âˆ’1, defining interval IÂ =Â [1,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]

(100, 123, 18320)-Net over F₅ — Digital

Digital (100, 123, 18320)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(100, 123, large)-Net in Base 5 — Upper bound on s

There is no (100, 123, large)-net in base 5, because

21 times m-reduction [i] would yield (100, 102, large)-net in base 5, but
- mutually orthogonal hypercube bound [i]