Best Known (19, 19+11, s)-Nets in Base 7
(19, 19+11, 128)-Net over F7 — Constructive and digital
Digital (19, 30, 128)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (3, 8, 28)-net over F7, using
- digital (11, 22, 100)-net over F7, using
- trace code for nets [i] based on digital (0, 11, 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, 11, 50)-net over F49, using
(19, 19+11, 355)-Net over F7 — Digital
Digital (19, 30, 355)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(730, 355, F7, 11) (dual of [355, 325, 12]-code), using
- 5 step Varšamov–Edel lengthening with (ri) = (2, 4 times 0) [i] based on linear OA(728, 348, F7, 11) (dual of [348, 320, 12]-code), using
- construction XX applied to C1 = C([341,8]), C2 = C([0,9]), C3 = C1 + C2 = C([0,8]), and C∩ = C1 ∩ C2 = C([341,9]) [i] based on
- linear OA(725, 342, F7, 10) (dual of [342, 317, 11]-code), using the primitive BCH-code C(I) with length 342 = 73−1, defining interval I = {−1,0,…,8}, and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(725, 342, F7, 10) (dual of [342, 317, 11]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 342 = 73−1, defining interval I = [0,9], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(728, 342, F7, 11) (dual of [342, 314, 12]-code), using the primitive BCH-code C(I) with length 342 = 73−1, defining interval I = {−1,0,…,9}, and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(722, 342, F7, 9) (dual of [342, 320, 10]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 342 = 73−1, defining interval I = [0,8], and designed minimum distance d ≥ |I|+1 = 10 [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([341,8]), C2 = C([0,9]), C3 = C1 + C2 = C([0,8]), and C∩ = C1 ∩ C2 = C([341,9]) [i] based on
- 5 step Varšamov–Edel lengthening with (ri) = (2, 4 times 0) [i] based on linear OA(728, 348, F7, 11) (dual of [348, 320, 12]-code), using
(19, 19+11, 34611)-Net in Base 7 — Upper bound on s
There is no (19, 30, 34612)-net in base 7, because
- 1 times m-reduction [i] would yield (19, 29, 34612)-net in base 7, but
- the generalized Rao bound for nets shows that 7m ≥ 3 220225 056302 247396 826969 > 729 [i]