Best Known (36−13, 36, s)-Nets in Base 7
(36−13, 36, 121)-Net over F7 — Constructive and digital
Digital (23, 36, 121)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (4, 10, 21)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (0, 3, 8)-net over F7, using
- net from sequence [i] based on digital (0, 7)-sequence over F7, using
- Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F7 with g(F) = 0 and N(F) ≥ 8, using
- the rational function field F7(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 7)-sequence over F7, using
- digital (1, 7, 13)-net over F7, using
- 6 times m-reduction [i] based on digital (1, 13, 13)-net over F7, using
- digital (0, 3, 8)-net over F7, using
- (u, u+v)-construction [i] based on
- 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, 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, 13, 50)-net over F49, using
- digital (4, 10, 21)-net over F7, using
(36−13, 36, 357)-Net over F7 — Digital
Digital (23, 36, 357)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(736, 357, F7, 13) (dual of [357, 321, 14]-code), using
- 4 step Varšamov–Edel lengthening with (ri) = (1, 0, 0, 0) [i] based on linear OA(735, 352, F7, 13) (dual of [352, 317, 14]-code), using
- construction XX applied to C1 = C([340,9]), C2 = C([0,10]), C3 = C1 + C2 = C([0,9]), and C∩ = C1 ∩ C2 = C([340,10]) [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 = {−2,−1,…,9}, and designed minimum distance d ≥ |I|+1 = 13 [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(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 = {−2,−1,…,10}, and designed minimum distance d ≥ |I|+1 = 14 [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(71, 7, F7, 1) (dual of [7, 6, 2]-code), using
- Reed–Solomon code RS(6,7) [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 XX applied to C1 = C([340,9]), C2 = C([0,10]), C3 = C1 + C2 = C([0,9]), and C∩ = C1 ∩ C2 = C([340,10]) [i] based on
- 4 step Varšamov–Edel lengthening with (ri) = (1, 0, 0, 0) [i] based on linear OA(735, 352, F7, 13) (dual of [352, 317, 14]-code), using
(36−13, 36, 42439)-Net in Base 7 — Upper bound on s
There is no (23, 36, 42440)-net in base 7, because
- 1 times m-reduction [i] would yield (23, 35, 42440)-net in base 7, but
- the generalized Rao bound for nets shows that 7m ≥ 378828 058305 117924 895475 186929 > 735 [i]