Best Known (56−21, 56, s)-Nets in Base 7
(56−21, 56, 116)-Net over F7 — Constructive and digital
Digital (35, 56, 116)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (4, 14, 16)-net over F7, using
- 2 times m-reduction [i] based on digital (4, 16, 16)-net over F7, using
- digital (21, 42, 100)-net over F7, using
- trace code for nets [i] based on digital (0, 21, 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, 21, 50)-net over F49, using
- digital (4, 14, 16)-net over F7, using
(56−21, 56, 353)-Net over F7 — Digital
Digital (35, 56, 353)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(756, 353, F7, 21) (dual of [353, 297, 22]-code), using
- 4 step Varšamov–Edel lengthening with (ri) = (1, 0, 0, 0) [i] based on linear OA(755, 348, F7, 21) (dual of [348, 293, 22]-code), using
- construction XX applied to C1 = C([341,18]), C2 = C([0,19]), C3 = C1 + C2 = C([0,18]), and C∩ = C1 ∩ C2 = C([341,19]) [i] based on
- 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 = {−1,0,…,18}, and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(752, 342, F7, 20) (dual of [342, 290, 21]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 342 = 73−1, defining interval I = [0,19], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(755, 342, F7, 21) (dual of [342, 287, 22]-code), using the primitive BCH-code C(I) with length 342 = 73−1, defining interval I = {−1,0,…,19}, and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(749, 342, F7, 19) (dual of [342, 293, 20]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 342 = 73−1, defining interval I = [0,18], and designed minimum distance d ≥ |I|+1 = 20 [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([341,18]), C2 = C([0,19]), C3 = C1 + C2 = C([0,18]), and C∩ = C1 ∩ C2 = C([341,19]) [i] based on
- 4 step Varšamov–Edel lengthening with (ri) = (1, 0, 0, 0) [i] based on linear OA(755, 348, F7, 21) (dual of [348, 293, 22]-code), using
(56−21, 56, 33557)-Net in Base 7 — Upper bound on s
There is no (35, 56, 33558)-net in base 7, because
- 1 times m-reduction [i] would yield (35, 55, 33558)-net in base 7, but
- the generalized Rao bound for nets shows that 7m ≥ 30234 870371 451110 321622 006029 341789 066116 083053 > 755 [i]