Best Known (27, 42, s)-Nets in Base 7
(27, 42, 126)-Net over F7 — Constructive and digital
Digital (27, 42, 126)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (5, 12, 26)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (1, 4, 50)-net over F7, using
- net defined by OOA [i] based on linear OOA(74, 50, F7, 3, 3) (dual of [(50, 3), 146, 4]-NRT-code), using
- digital (1, 8, 13)-net over F7, using
- 5 times m-reduction [i] based on digital (1, 13, 13)-net over F7, using
- digital (1, 4, 50)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (15, 30, 100)-net over F7, using
- trace code for nets [i] based on digital (0, 15, 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, 15, 50)-net over F49, using
- digital (5, 12, 26)-net over F7, using
(27, 42, 379)-Net over F7 — Digital
Digital (27, 42, 379)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(742, 379, F7, 15) (dual of [379, 337, 16]-code), using
- 28 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 6 times 0, 1, 17 times 0) [i] based on linear OA(738, 347, F7, 15) (dual of [347, 309, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(12) [i] based on
- linear OA(737, 343, F7, 15) (dual of [343, 306, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 342 = 73−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(734, 343, F7, 13) (dual of [343, 309, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 342 = 73−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(71, 4, F7, 1) (dual of [4, 3, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(71, s, F7, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(14) ⊂ Ce(12) [i] based on
- 28 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 6 times 0, 1, 17 times 0) [i] based on linear OA(738, 347, F7, 15) (dual of [347, 309, 16]-code), using
(27, 42, 50187)-Net in Base 7 — Upper bound on s
There is no (27, 42, 50188)-net in base 7, because
- 1 times m-reduction [i] would yield (27, 41, 50188)-net in base 7, but
- the generalized Rao bound for nets shows that 7m ≥ 44573 815708 188057 800567 824964 143777 > 741 [i]