MinT - Best Known (29, 29+13, s)-Nets in Base 5

Best Known (29, 29+13, s)-Nets in Base 5

(29, 29+13, 146)-Net over F₅ — Constructive and digital

Digital (29, 42, 146)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (2, 8, 14)-net over F₅, using
  - linear code embedding found in a computer search [i]
2. digital (21, 34, 132)-net over F₅, using
  - trace code for nets [i] based on digital (4, 17, 66)-net over F₂₅, using
    - net from sequence [i] based on digital (4, 65)-sequence over F₂₅, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 4 and N(F) ≥ 66, using
        a function field by GarcÃa and Quoos [i]

(29, 29+13, 488)-Net over F₅ — Digital

Digital (29, 42, 488)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5⁴², 488, F₅, 13) (dual of [488, 446, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(5⁴², 635, F₅, 13) (dual of [635, 593, 14]-code), using
  - constructionÂ X applied to C([0,6]) âŠ‚ C([0,5]) [i] based on
    1. linear OA(5⁴¹, 626, F₅, 13) (dual of [626, 585, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 626Â | 5⁸âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]
    2. linear OA(5³³, 626, F₅, 11) (dual of [626, 593, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 626Â | 5⁸âˆ’1, defining interval IÂ =Â [0,5], and minimum distance dÂ â‰¥ |{−5,−4,…,5}|+1Â =Â 12 (BCH-bound) [i]
    3. linear OA(5¹, 9, F₅, 1) (dual of [9, 8, 2]-code), using
      - discarding factors / shortening the dual code based on linear OA(5¹, s, F₅, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(29, 29+13, 44711)-Net in Base 5 — Upper bound on s

There is no (29, 42, 44712)-net in base 5, because

1 times m-reduction [i] would yield (29, 41, 44712)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 45478 394897 576432 748962 288257 > 5⁴¹ [i]