MinT - Best Known (142−21, 142, s)-Nets in Base 5

Best Known (142−21, 142, s)-Nets in Base 5

(142−21, 142, 39078)-Net over F₅ — Constructive and digital

Digital (121, 142, 39078)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (3, 13, 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 (108, 129, 39062)-net over F₅, using
  - net defined by OOA [i] based on linear OOA(5¹²⁹, 39062, F₅, 21, 21) (dual of [(39062, 21), 820173, 22]-NRT-code), using
    - OOA 10-folding and stacking with additional row [i] based on linear OA(5¹²⁹, 390621, F₅, 21) (dual of [390621, 390492, 22]-code), using
      - discarding factors / shortening the dual code based on linear OA(5¹²⁹, 390625, F₅, 21) (dual of [390625, 390496, 22]-code), using
        an extension C_e(20) of the primitive narrow-sense BCH-code C(I) with length 390624Â = 5⁸âˆ’1, defining interval IÂ =Â [1,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]

(142−21, 142, 304951)-Net over F₅ — Digital

Digital (121, 142, 304951)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5¹⁴², 304951, F₅, 21) (dual of [304951, 304809, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(5¹⁴², 390643, F₅, 21) (dual of [390643, 390501, 22]-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OA(5¹³, 17, F₅, 10) (dual of [17, 4, 11]-code), using
      - 9 times truncation [i] based on linear OA(5²², 26, F₅, 19) (dual of [26, 4, 20]-code), using
        projective code from ovoid in PG(3,Â 5) [i]
    2. linear OA(5¹²⁹, 390626, F₅, 21) (dual of [390626, 390497, 22]-code), using
      - the expurgated narrow-sense BCH-code C(I) with length 390626Â | 5¹⁶âˆ’1, defining interval IÂ =Â [0,10], and minimum distance dÂ â‰¥ |{−10,−9,…,10}|+1Â =Â 22 (BCH-bound) [i]

(142−21, 142, large)-Net in Base 5 — Upper bound on s

There is no (121, 142, large)-net in base 5, because

19 times m-reduction [i] would yield (121, 123, large)-net in base 5, but
- mutually orthogonal hypercube bound [i]