Best Known (53, 72, s)-Nets in Base 7
(53, 72, 308)-Net over F7 — Constructive and digital
Digital (53, 72, 308)-net over F7, using
- generalized (u, u+v)-construction [i] based on
- digital (0, 4, 8)-net over F7, using
- net from sequence [i] based on digital (0, 7)-sequence over F7, using
- Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F7 with g(F) = 0 and N(F) ≥ 8, using
- the rational function field F7(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 7)-sequence over F7, using
- 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 (9, 18, 100)-net over F7, using
- trace code for nets [i] based on digital (0, 9, 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, 9, 50)-net over F49, using
- digital (19, 38, 100)-net over F7, using
- trace code for nets [i] based on digital (0, 19, 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, 19, 50)-net over F49, using
- digital (0, 4, 8)-net over F7, using
(53, 72, 3045)-Net over F7 — Digital
Digital (53, 72, 3045)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(772, 3045, F7, 19) (dual of [3045, 2973, 20]-code), using
- 633 step Varšamov–Edel lengthening with (ri) = (2, 4 times 0, 1, 16 times 0, 1, 45 times 0, 1, 102 times 0, 1, 188 times 0, 1, 272 times 0) [i] based on linear OA(765, 2405, F7, 19) (dual of [2405, 2340, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(17) [i] based on
- 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(761, 2401, F7, 18) (dual of [2401, 2340, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [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(18) ⊂ Ce(17) [i] based on
- 633 step Varšamov–Edel lengthening with (ri) = (2, 4 times 0, 1, 16 times 0, 1, 45 times 0, 1, 102 times 0, 1, 188 times 0, 1, 272 times 0) [i] based on linear OA(765, 2405, F7, 19) (dual of [2405, 2340, 20]-code), using
(53, 72, 3209844)-Net in Base 7 — Upper bound on s
There is no (53, 72, 3209845)-net in base 7, because
- 1 times m-reduction [i] would yield (53, 71, 3209845)-net in base 7, but
- the generalized Rao bound for nets shows that 7m ≥ 1 004526 665170 647068 256089 424693 416083 784012 466262 884387 090175 > 771 [i]