MinT - Best Known (122, 122+28, s)-Nets in Base 5

Best Known (122, 122+28, s)-Nets in Base 5

(122, 122+28, 1132)-Net over F₅ — Constructive and digital

Digital (122, 150, 1132)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (3, 17, 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 (105, 133, 1116)-net over F₅, using
  - net defined by OOA [i] based on linear OOA(5¹³³, 1116, F₅, 28, 28) (dual of [(1116, 28), 31115, 29]-NRT-code), using
    - OA 14-folding and stacking [i] based on linear OA(5¹³³, 15624, F₅, 28) (dual of [15624, 15491, 29]-code), using
      - discarding factors / shortening the dual code based on linear OA(5¹³³, 15625, F₅, 28) (dual of [15625, 15492, 29]-code), using
        an extension C_e(27) of the primitive narrow-sense BCH-code C(I) with length 15624Â = 5⁶âˆ’1, defining interval IÂ =Â [1,27], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 28 [i]

(122, 122+28, 20883)-Net over F₅ — Digital

Digital (122, 150, 20883)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(122, 122+28, large)-Net in Base 5 — Upper bound on s

There is no (122, 150, large)-net in base 5, because

26 times m-reduction [i] would yield (122, 124, large)-net in base 5, but
- mutually orthogonal hypercube bound [i]