Best Known (57, 78, s)-Nets in Base 3
(57, 78, 204)-Net over F3 — Constructive and digital
Digital (57, 78, 204)-net over F3, using
- trace code for nets [i] based on digital (5, 26, 68)-net over F27, using
- net from sequence [i] based on digital (5, 67)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 5 and N(F) ≥ 68, using
- net from sequence [i] based on digital (5, 67)-sequence over F27, using
(57, 78, 312)-Net over F3 — Digital
Digital (57, 78, 312)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(378, 312, F3, 21) (dual of [312, 234, 22]-code), using
- 62 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, 1, 14 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]
- 62 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, 1, 14 times 0) [i] based on linear OA(370, 242, F3, 21) (dual of [242, 172, 22]-code), using
(57, 78, 10675)-Net in Base 3 — Upper bound on s
There is no (57, 78, 10676)-net in base 3, because
- 1 times m-reduction [i] would yield (57, 77, 10676)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 5 476258 878289 420319 138631 760417 170649 > 377 [i]