Best Known (21, 21+12, s)-Nets in Base 7
(21, 21+12, 116)-Net over F7 — Constructive and digital
Digital (21, 33, 116)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (3, 9, 16)-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 (0, 6, 8)-net over F7, using
- net from sequence [i] based on digital (0, 7)-sequence over F7 (see above)
- digital (0, 3, 8)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (12, 24, 100)-net over F7, using
- trace code for nets [i] based on digital (0, 12, 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, 12, 50)-net over F49, using
- digital (3, 9, 16)-net over F7, using
(21, 21+12, 356)-Net over F7 — Digital
Digital (21, 33, 356)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(733, 356, F7, 12) (dual of [356, 323, 13]-code), using
- construction XX applied to C1 = C([340,7]), C2 = C([0,9]), C3 = C1 + C2 = C([0,7]), and C∩ = C1 ∩ C2 = C([340,9]) [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 = {−2,−1,…,7}, 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(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(719, 342, F7, 8) (dual of [342, 323, 9]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 342 = 73−1, defining interval I = [0,7], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(71, 7, F7, 1) (dual of [7, 6, 2]-code), using
- Reed–Solomon code RS(6,7) [i]
- linear OA(71, 7, F7, 1) (dual of [7, 6, 2]-code) (see above)
- construction XX applied to C1 = C([340,7]), C2 = C([0,9]), C3 = C1 + C2 = C([0,7]), and C∩ = C1 ∩ C2 = C([340,9]) [i] based on
(21, 21+12, 22184)-Net in Base 7 — Upper bound on s
There is no (21, 33, 22185)-net in base 7, because
- the generalized Rao bound for nets shows that 7m ≥ 7732 904477 354144 172147 369313 > 733 [i]