Best Known (18−6, 18, s)-Nets in Base 7
(18−6, 18, 200)-Net over F7 — Constructive and digital
Digital (12, 18, 200)-net over F7, using
- 1 times m-reduction [i] based on digital (12, 19, 200)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (2, 5, 132)-net over F7, using
- net defined by OOA [i] based on linear OOA(75, 132, F7, 3, 3) (dual of [(132, 3), 391, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(75, 132, F7, 2, 3) (dual of [(132, 2), 259, 4]-NRT-code), using
- net defined by OOA [i] based on linear OOA(75, 132, F7, 3, 3) (dual of [(132, 3), 391, 4]-NRT-code), using
- digital (7, 14, 100)-net over F7, using
- trace code for nets [i] based on digital (0, 7, 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, 7, 50)-net over F49, using
- digital (2, 5, 132)-net over F7, using
- (u, u+v)-construction [i] based on
(18−6, 18, 497)-Net over F7 — Digital
Digital (12, 18, 497)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(718, 497, F7, 6) (dual of [497, 479, 7]-code), using
- 147 step Varšamov–Edel lengthening with (ri) = (1, 34 times 0, 1, 111 times 0) [i] based on linear OA(716, 348, F7, 6) (dual of [348, 332, 7]-code), using
- construction XX applied to C1 = C([341,3]), C2 = C([0,4]), C3 = C1 + C2 = C([0,3]), and C∩ = C1 ∩ C2 = C([341,4]) [i] based on
- linear OA(713, 342, F7, 5) (dual of [342, 329, 6]-code), using the primitive BCH-code C(I) with length 342 = 73−1, defining interval I = {−1,0,1,2,3}, and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(713, 342, F7, 5) (dual of [342, 329, 6]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 342 = 73−1, defining interval I = [0,4], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(716, 342, F7, 6) (dual of [342, 326, 7]-code), using the primitive BCH-code C(I) with length 342 = 73−1, defining interval I = {−1,0,…,4}, and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(710, 342, F7, 4) (dual of [342, 332, 5]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 342 = 73−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 5 [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
- linear OA(70, 3, F7, 0) (dual of [3, 3, 1]-code) (see above)
- construction XX applied to C1 = C([341,3]), C2 = C([0,4]), C3 = C1 + C2 = C([0,3]), and C∩ = C1 ∩ C2 = C([341,4]) [i] based on
- 147 step Varšamov–Edel lengthening with (ri) = (1, 34 times 0, 1, 111 times 0) [i] based on linear OA(716, 348, F7, 6) (dual of [348, 332, 7]-code), using
(18−6, 18, 35628)-Net in Base 7 — Upper bound on s
There is no (12, 18, 35629)-net in base 7, because
- the generalized Rao bound for nets shows that 7m ≥ 1628 426833 959223 > 718 [i]