MinT - Best Known (40−17, 40, s)-Nets in Base 49

Best Known (40−17, 40, s)-Nets in Base 49

(40−17, 40, 344)-Net over F₄₉ — Constructive and digital

Digital (23, 40, 344)-net over F₄₉, using

t-expansion [i] based on digital (21, 40, 344)-net over F₄₉, using
- net from sequence [i] based on digital (21, 343)-sequence over F₄₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄₉ with g(F) = 21 and N(F) ≥ 344, using
    - the Hermitian function field over F₄₉ [i]

(40−17, 40, 2677)-Net over F₄₉ — Digital

Digital (23, 40, 2677)-net over F₄₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(49⁴⁰, 2677, F₄₉, 17) (dual of [2677, 2637, 18]-code), using
- 267 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (4, 0, 0, 0, 1, 16 times 0, 1, 59 times 0, 1, 185 times 0) [i] based on linear OA(49³³, 2403, F₄₉, 17) (dual of [2403, 2370, 18]-code), using
  - constructionÂ X applied to C_e(16) âŠ‚ C_e(15) [i] based on
    1. linear OA(49³³, 2401, F₄₉, 17) (dual of [2401, 2368, 18]-code), using an extension C_e(16) of the primitive narrow-sense BCH-code C(I) with length 2400Â = 49²âˆ’1, defining interval IÂ =Â [1,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]
    2. linear OA(49³¹, 2401, F₄₉, 16) (dual of [2401, 2370, 17]-code), using an extension C_e(15) of the primitive narrow-sense BCH-code C(I) with length 2400Â = 49²âˆ’1, defining interval IÂ =Â [1,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]
    3. linear OA(49⁰, 2, F₄₉, 0) (dual of [2, 2, 1]-code), using
      - discarding factors / shortening the dual code based on linear OA(49⁰, s, F₄₉, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(40−17, 40, large)-Net in Base 49 — Upper bound on s

There is no (23, 40, large)-net in base 49, because

15 times m-reduction [i] would yield (23, 25, large)-net in base 49, but
- mutually orthogonal hypercube bound [i]