Best Known (27−9, 27, s)-Nets in Base 7
(27−9, 27, 200)-Net over F7 — Constructive and digital
Digital (18, 27, 200)-net over F7, using
- 71 times duplication [i] based on digital (17, 26, 200)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (4, 8, 100)-net over F7, using
- trace code for nets [i] based on digital (0, 4, 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, 4, 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 (4, 8, 100)-net over F7, using
- (u, u+v)-construction [i] based on
(27−9, 27, 471)-Net over F7 — Digital
Digital (18, 27, 471)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(727, 471, F7, 9) (dual of [471, 444, 10]-code), using
- 120 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 0, 1, 13 times 0, 1, 33 times 0, 1, 67 times 0) [i] based on linear OA(722, 346, F7, 9) (dual of [346, 324, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(7) [i] based on
- linear OA(722, 343, F7, 9) (dual of [343, 321, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 342 = 73−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(719, 343, F7, 8) (dual of [343, 324, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 342 = 73−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(70, 3, F7, 0) (dual of [3, 3, 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(8) ⊂ Ce(7) [i] based on
- 120 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 0, 1, 13 times 0, 1, 33 times 0, 1, 67 times 0) [i] based on linear OA(722, 346, F7, 9) (dual of [346, 324, 10]-code), using
(27−9, 27, 114823)-Net in Base 7 — Upper bound on s
There is no (18, 27, 114824)-net in base 7, because
- 1 times m-reduction [i] would yield (18, 26, 114824)-net in base 7, but
- the generalized Rao bound for nets shows that 7m ≥ 9387 657281 191897 603777 > 726 [i]