Best Known (52−18, 52, s)-Nets in Base 7
(52−18, 52, 130)-Net over F7 — Constructive and digital
Digital (34, 52, 130)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (7, 16, 30)-net over F7, using
- digital (18, 36, 100)-net over F7, using
- trace code for nets [i] based on digital (0, 18, 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, 18, 50)-net over F49, using
(52−18, 52, 470)-Net over F7 — Digital
Digital (34, 52, 470)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(752, 470, F7, 18) (dual of [470, 418, 19]-code), using
- 116 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 0, 1, 9 times 0, 1, 20 times 0, 1, 34 times 0, 1, 45 times 0) [i] based on linear OA(746, 348, F7, 18) (dual of [348, 302, 19]-code), using
- construction XX applied to C1 = C([341,15]), C2 = C([0,16]), C3 = C1 + C2 = C([0,15]), and C∩ = C1 ∩ C2 = C([341,16]) [i] based on
- linear OA(743, 342, F7, 17) (dual of [342, 299, 18]-code), using the primitive BCH-code C(I) with length 342 = 73−1, defining interval I = {−1,0,…,15}, and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(743, 342, F7, 17) (dual of [342, 299, 18]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 342 = 73−1, defining interval I = [0,16], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(746, 342, F7, 18) (dual of [342, 296, 19]-code), using the primitive BCH-code C(I) with length 342 = 73−1, defining interval I = {−1,0,…,16}, and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(740, 342, F7, 16) (dual of [342, 302, 17]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 342 = 73−1, defining interval I = [0,15], and designed minimum distance d ≥ |I|+1 = 17 [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([341,15]), C2 = C([0,16]), C3 = C1 + C2 = C([0,15]), and C∩ = C1 ∩ C2 = C([341,16]) [i] based on
- 116 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 0, 1, 9 times 0, 1, 20 times 0, 1, 34 times 0, 1, 45 times 0) [i] based on linear OA(746, 348, F7, 18) (dual of [348, 302, 19]-code), using
(52−18, 52, 52764)-Net in Base 7 — Upper bound on s
There is no (34, 52, 52765)-net in base 7, because
- the generalized Rao bound for nets shows that 7m ≥ 88 129049 931348 666820 043767 063833 596926 893679 > 752 [i]