Best Known (56, 56+21, s)-Nets in Base 3
(56, 56+21, 192)-Net over F3 — Constructive and digital
Digital (56, 77, 192)-net over F3, using
- 1 times m-reduction [i] based on digital (56, 78, 192)-net over F3, using
- trace code for nets [i] based on digital (4, 26, 64)-net over F27, using
- net from sequence [i] based on digital (4, 63)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 4 and N(F) ≥ 64, using
- net from sequence [i] based on digital (4, 63)-sequence over F27, using
- trace code for nets [i] based on digital (4, 26, 64)-net over F27, using
(56, 56+21, 296)-Net over F3 — Digital
Digital (56, 77, 296)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(377, 296, F3, 21) (dual of [296, 219, 22]-code), using
- 47 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 0, 0, 0, 1, 6 times 0, 1, 8 times 0, 1, 11 times 0, 1, 12 times 0) [i] based on linear OA(370, 242, F3, 21) (dual of [242, 172, 22]-code), using
- the primitive narrow-sense BCH-code C(I) with length 242 = 35−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- 47 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 0, 0, 0, 1, 6 times 0, 1, 8 times 0, 1, 11 times 0, 1, 12 times 0) [i] based on linear OA(370, 242, F3, 21) (dual of [242, 172, 22]-code), using
(56, 56+21, 9563)-Net in Base 3 — Upper bound on s
There is no (56, 77, 9564)-net in base 3, because
- 1 times m-reduction [i] would yield (56, 76, 9564)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 1 824914 601852 823623 843515 923780 949577 > 376 [i]