MinT - Best Known (136−25, 136, s)-Nets in Base 5

Best Known (136−25, 136, s)-Nets in Base 5

(136−25, 136, 1319)-Net over F₅ — Constructive and digital

Digital (111, 136, 1319)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (4, 16, 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 (95, 120, 1301)-net over F₅, using
  - net defined by OOA [i] based on linear OOA(5¹²⁰, 1301, F₅, 25, 25) (dual of [(1301, 25), 32405, 26]-NRT-code), using
    - OOA 12-folding and stacking with additional row [i] based on linear OA(5¹²⁰, 15613, F₅, 25) (dual of [15613, 15493, 26]-code), using
      - discarding factors / shortening the dual code based on linear OA(5¹²⁰, 15624, F₅, 25) (dual of [15624, 15504, 26]-code), using
        1 times truncation [i] based on linear OA(5¹²¹, 15625, F₅, 26) (dual of [15625, 15504, 27]-code), using
        an extension C_e(25) of the primitive narrow-sense BCH-code C(I) with length 15624Â = 5⁶âˆ’1, defining interval IÂ =Â [1,25], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 26 [i]

(136−25, 136, 22406)-Net over F₅ — Digital

Digital (111, 136, 22406)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(136−25, 136, large)-Net in Base 5 — Upper bound on s

There is no (111, 136, large)-net in base 5, because

23 times m-reduction [i] would yield (111, 113, large)-net in base 5, but
- mutually orthogonal hypercube bound [i]