Best Known (51, 71, s)-Nets in Base 7
(51, 71, 250)-Net over F7 — Constructive and digital
Digital (51, 71, 250)-net over F7, using
- generalized (u, u+v)-construction [i] based on
- digital (5, 11, 50)-net over F7, using
- base reduction for projective spaces (embedding PG(5,49) in PG(10,7)) 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
- base reduction for projective spaces (embedding PG(5,49) in PG(10,7)) 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 (5, 11, 50)-net over F7, using
(51, 71, 2413)-Net over F7 — Digital
Digital (51, 71, 2413)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(771, 2413, F7, 20) (dual of [2413, 2342, 21]-code), using
- 6 step Varšamov–Edel lengthening with (ri) = (2, 5 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
- 6 step Varšamov–Edel lengthening with (ri) = (2, 5 times 0) [i] based on linear OA(769, 2405, F7, 20) (dual of [2405, 2336, 21]-code), using
(51, 71, 755122)-Net in Base 7 — Upper bound on s
There is no (51, 71, 755123)-net in base 7, because
- the generalized Rao bound for nets shows that 7m ≥ 1 004527 711347 240329 443749 112493 371125 418883 924333 362053 691501 > 771 [i]