Best Known (54, 54+21, s)-Nets in Base 7
(54, 54+21, 250)-Net over F7 — Constructive and digital
Digital (54, 75, 250)-net over F7, using
- generalized (u, u+v)-construction [i] based on
- digital (6, 13, 50)-net over F7, using
- base reduction for projective spaces (embedding PG(6,49) in PG(12,7)) for nets [i] based on digital (0, 7, 50)-net over F49, using
- net from sequence [i] based on digital (0, 49)-sequence over F49, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F49 with g(F) = 0 and N(F) ≥ 50, using
- the rational function field F49(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 49)-sequence over F49, using
- base reduction for projective spaces (embedding PG(6,49) in PG(12,7)) for nets [i] based on digital (0, 7, 50)-net over F49, using
- digital (10, 20, 100)-net over F7, using
- trace code for nets [i] based on digital (0, 10, 50)-net over F49, using
- net from sequence [i] based on digital (0, 49)-sequence over F49 (see above)
- trace code for nets [i] based on digital (0, 10, 50)-net over F49, using
- digital (21, 42, 100)-net over F7, using
- trace code for nets [i] based on digital (0, 21, 50)-net over F49, using
- net from sequence [i] based on digital (0, 49)-sequence over F49 (see above)
- trace code for nets [i] based on digital (0, 21, 50)-net over F49, using
- digital (6, 13, 50)-net over F7, using
(54, 54+21, 2427)-Net over F7 — Digital
Digital (54, 75, 2427)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(775, 2427, F7, 21) (dual of [2427, 2352, 22]-code), using
- 24 step Varšamov–Edel lengthening with (ri) = (2, 4 times 0, 1, 18 times 0) [i] based on linear OA(772, 2400, F7, 21) (dual of [2400, 2328, 22]-code), using
- 1 times truncation [i] based on linear OA(773, 2401, F7, 22) (dual of [2401, 2328, 23]-code), using
- an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- 1 times truncation [i] based on linear OA(773, 2401, F7, 22) (dual of [2401, 2328, 23]-code), using
- 24 step Varšamov–Edel lengthening with (ri) = (2, 4 times 0, 1, 18 times 0) [i] based on linear OA(772, 2400, F7, 21) (dual of [2400, 2328, 22]-code), using
(54, 54+21, 1353781)-Net in Base 7 — Upper bound on s
There is no (54, 75, 1353782)-net in base 7, because
- 1 times m-reduction [i] would yield (54, 74, 1353782)-net in base 7, but
- the generalized Rao bound for nets shows that 7m ≥ 344 553453 645381 001495 548325 795718 578325 525807 648795 600173 040173 > 774 [i]