Best Known (73, 73+30, s)-Nets in Base 7
(73, 73+30, 688)-Net over F7 — Constructive and digital
Digital (73, 103, 688)-net over F7, using
- 1 times m-reduction [i] based on digital (73, 104, 688)-net over F7, using
- trace code for nets [i] based on digital (21, 52, 344)-net over F49, using
- net from sequence [i] based on digital (21, 343)-sequence over F49, using
- Niederreiter–Xing sequence construction 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]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F49 with g(F) = 21 and N(F) ≥ 344, using
- net from sequence [i] based on digital (21, 343)-sequence over F49, using
- trace code for nets [i] based on digital (21, 52, 344)-net over F49, using
(73, 73+30, 2239)-Net over F7 — Digital
Digital (73, 103, 2239)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(7103, 2239, F7, 30) (dual of [2239, 2136, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(7103, 2409, F7, 30) (dual of [2409, 2306, 31]-code), using
- construction X applied to Ce(29) ⊂ Ce(26) [i] based on
- linear OA(7101, 2401, F7, 30) (dual of [2401, 2300, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(793, 2401, F7, 27) (dual of [2401, 2308, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(72, 8, F7, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,7)), using
- extended Reed–Solomon code RSe(6,7) [i]
- Hamming code H(2,7) [i]
- algebraic-geometric code AG(F, Q+1P) with degQ = 3 and degPÂ =Â 2 [i] based on function field F/F7 with g(F) = 0 and N(F) ≥ 8, using the rational function field F7(x) [i]
- construction X applied to Ce(29) ⊂ Ce(26) [i] based on
- discarding factors / shortening the dual code based on linear OA(7103, 2409, F7, 30) (dual of [2409, 2306, 31]-code), using
(73, 73+30, 680167)-Net in Base 7 — Upper bound on s
There is no (73, 103, 680168)-net in base 7, because
- the generalized Rao bound for nets shows that 7m ≥ 1109 438077 504059 842629 703875 162319 333629 995516 177716 851815 533195 406093 860490 905491 440193 > 7103 [i]