Best Known (35, 55, s)-Nets in Base 7
(35, 55, 118)-Net over F7 — Constructive and digital
Digital (35, 55, 118)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (5, 15, 18)-net over F7, using
- 3 times m-reduction [i] based on digital (5, 18, 18)-net over F7, using
- digital (20, 40, 100)-net over F7, using
- trace code for nets [i] based on digital (0, 20, 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, 20, 50)-net over F49, using
- digital (5, 15, 18)-net over F7, using
(35, 55, 383)-Net over F7 — Digital
Digital (35, 55, 383)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(755, 383, F7, 20) (dual of [383, 328, 21]-code), using
- 29 step Varšamov–Edel lengthening with (ri) = (1, 6 times 0, 1, 21 times 0) [i] based on linear OA(753, 352, F7, 20) (dual of [352, 299, 21]-code), using
- construction XX applied to C1 = C([340,16]), C2 = C([0,17]), C3 = C1 + C2 = C([0,16]), and C∩ = C1 ∩ C2 = C([340,17]) [i] based on
- linear OA(749, 342, F7, 19) (dual of [342, 293, 20]-code), using the primitive BCH-code C(I) with length 342 = 73−1, defining interval I = {−2,−1,…,16}, and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(746, 342, F7, 18) (dual of [342, 296, 19]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 342 = 73−1, defining interval I = [0,17], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(752, 342, F7, 20) (dual of [342, 290, 21]-code), using the primitive BCH-code C(I) with length 342 = 73−1, defining interval I = {−2,−1,…,17}, and designed minimum distance d ≥ |I|+1 = 21 [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(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([340,16]), C2 = C([0,17]), C3 = C1 + C2 = C([0,16]), and C∩ = C1 ∩ C2 = C([340,17]) [i] based on
- 29 step Varšamov–Edel lengthening with (ri) = (1, 6 times 0, 1, 21 times 0) [i] based on linear OA(753, 352, F7, 20) (dual of [352, 299, 21]-code), using
(35, 55, 33557)-Net in Base 7 — Upper bound on s
There is no (35, 55, 33558)-net in base 7, because
- the generalized Rao bound for nets shows that 7m ≥ 30234 870371 451110 321622 006029 341789 066116 083053 > 755 [i]