Best Known (26, 26+13, s)-Nets in Base 7
(26, 26+13, 200)-Net over F7 — Constructive and digital
Digital (26, 39, 200)-net over F7, using
- 71 times duplication [i] based on digital (25, 38, 200)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (6, 12, 100)-net over F7, using
- trace code 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
- trace code for nets [i] based on digital (0, 6, 50)-net over F49, using
- digital (13, 26, 100)-net over F7, using
- trace code for nets [i] based on digital (0, 13, 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, 13, 50)-net over F49, using
- digital (6, 12, 100)-net over F7, using
- (u, u+v)-construction [i] based on
(26, 26+13, 499)-Net over F7 — Digital
Digital (26, 39, 499)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(739, 499, F7, 13) (dual of [499, 460, 14]-code), using
- 146 step Varšamov–Edel lengthening with (ri) = (2, 5 times 0, 1, 22 times 0, 1, 47 times 0, 1, 68 times 0) [i] based on linear OA(734, 348, F7, 13) (dual of [348, 314, 14]-code), using
- construction XX applied to C1 = C([341,10]), C2 = C([0,11]), C3 = C1 + C2 = C([0,10]), and C∩ = C1 ∩ C2 = C([341,11]) [i] based on
- linear OA(731, 342, F7, 12) (dual of [342, 311, 13]-code), using the primitive BCH-code C(I) with length 342 = 73−1, defining interval I = {−1,0,…,10}, and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(731, 342, F7, 12) (dual of [342, 311, 13]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 342 = 73−1, defining interval I = [0,11], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(734, 342, F7, 13) (dual of [342, 308, 14]-code), using the primitive BCH-code C(I) with length 342 = 73−1, defining interval I = {−1,0,…,11}, and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(728, 342, F7, 11) (dual of [342, 314, 12]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 342 = 73−1, defining interval I = [0,10], and designed minimum distance d ≥ |I|+1 = 12 [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([341,10]), C2 = C([0,11]), C3 = C1 + C2 = C([0,10]), and C∩ = C1 ∩ C2 = C([341,11]) [i] based on
- 146 step Varšamov–Edel lengthening with (ri) = (2, 5 times 0, 1, 22 times 0, 1, 47 times 0, 1, 68 times 0) [i] based on linear OA(734, 348, F7, 13) (dual of [348, 314, 14]-code), using
(26, 26+13, 112290)-Net in Base 7 — Upper bound on s
There is no (26, 39, 112291)-net in base 7, because
- 1 times m-reduction [i] would yield (26, 38, 112291)-net in base 7, but
- the generalized Rao bound for nets shows that 7m ≥ 129 934930 804537 894995 047469 139045 > 738 [i]