MinT - Best Known (111, 111+26, s)-Nets in Base 5

Best Known (111, 111+26, s)-Nets in Base 5

(111, 111+26, 1217)-Net over F₅ — Constructive and digital

Digital (111, 137, 1217)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (3, 16, 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 (95, 121, 1201)-net over F₅, using
  - net defined by OOA [i] based on linear OOA(5¹²¹, 1201, F₅, 26, 26) (dual of [(1201, 26), 31105, 27]-NRT-code), using
    - OA 13-folding and stacking [i] based on linear OA(5¹²¹, 15613, F₅, 26) (dual of [15613, 15492, 27]-code), using
      - discarding factors / shortening the dual code 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]

(111, 111+26, 17228)-Net over F₅ — Digital

Digital (111, 137, 17228)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(111, 111+26, large)-Net in Base 5 — Upper bound on s

There is no (111, 137, large)-net in base 5, because

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