MinT - Best Known (13, 13+19, s)-Nets in Base 49

Best Known (13, 13+19, s)-Nets in Base 49

(13, 13+19, 104)-Net over F₄₉ — Constructive and digital

Digital (13, 32, 104)-net over F₄₉, using

(u,Â u+v)-construction [i] based on
1. digital (2, 11, 52)-net over F₄₉, using
  - net from sequence [i] based on digital (2, 51)-sequence over F₄₉, using
    - Niederreiter sequence [i]
2. digital (2, 21, 52)-net over F₄₉, using
  - net from sequence [i] based on digital (2, 51)-sequence over F₄₉ (see above)

(13, 13+19, 169)-Net over F₄₉ — Digital

Digital (13, 32, 169)-net over F₄₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(49³², 169, F₄₉, 19) (dual of [169, 137, 20]-code), using
- 66 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (5, 0, 1, 4 times 0, 1, 10 times 0, 1, 18 times 0, 1, 28 times 0) [i] based on linear OA(49²³, 94, F₄₉, 19) (dual of [94, 71, 20]-code), using
  - constructionÂ X applied to AG(F,71P) âŠ‚ AG(F,73P) [i] based on
    1. linear OA(49²², 91, F₄₉, 19) (dual of [91, 69, 20]-code), using algebraic-geometric code AG(F,71P) [i] based on function field F/F₄₉ with g(F) = 3 and N(F) ≥ 92, using
      - a curve from the manYPoints database [i]
    2. linear OA(49²⁰, 91, F₄₉, 17) (dual of [91, 71, 18]-code), using algebraic-geometric code AG(F,73P) [i] based on function field F/F₄₉ with g(F) = 3 and N(F) ≥ 92 (see above)
    3. linear OA(49¹, 3, F₄₉, 1) (dual of [3, 2, 2]-code), using
      - discarding factors / shortening the dual code based on linear OA(49¹, s, F₄₉, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(13, 13+19, 57314)-Net in Base 49 — Upper bound on s

There is no (13, 32, 57315)-net in base 49, because

1 times m-reduction [i] would yield (13, 31, 57315)-net in base 49, but
- the generalized Rao bound for nets shows that 49^mÂ â‰¥ 24894 782104 298405 406740 993514 868058 679601 661391 611665 > 49³¹ [i]