Best Known (52, 72, s)-Nets in Base 7
(52, 72, 300)-Net over F7 — Constructive and digital
Digital (52, 72, 300)-net over F7, using
- generalized (u, u+v)-construction [i] based on
- digital (6, 12, 100)-net over F7, using
- trace code for nets [i] based on digital (0, 6, 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
- trace code for nets [i] based on digital (0, 6, 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 (20, 40, 100)-net over F7, using
- trace code for nets [i] based on digital (0, 20, 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, 20, 50)-net over F49, using
- digital (6, 12, 100)-net over F7, using
(52, 72, 2440)-Net over F7 — Digital
Digital (52, 72, 2440)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(772, 2440, F7, 20) (dual of [2440, 2368, 21]-code), using
- 32 step Varšamov–Edel lengthening with (ri) = (2, 5 times 0, 1, 25 times 0) [i] based on linear OA(769, 2405, F7, 20) (dual of [2405, 2336, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(18) [i] based on
- linear OA(769, 2401, F7, 20) (dual of [2401, 2332, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(765, 2401, F7, 19) (dual of [2401, 2336, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(70, 4, F7, 0) (dual of [4, 4, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(70, s, F7, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(19) ⊂ Ce(18) [i] based on
- 32 step Varšamov–Edel lengthening with (ri) = (2, 5 times 0, 1, 25 times 0) [i] based on linear OA(769, 2405, F7, 20) (dual of [2405, 2336, 21]-code), using
(52, 72, 917335)-Net in Base 7 — Upper bound on s
There is no (52, 72, 917336)-net in base 7, because
- the generalized Rao bound for nets shows that 7m ≥ 7 031743 799654 610559 943125 015469 860260 931158 186412 896573 947473 > 772 [i]