Best Known (76−21, 76, s)-Nets in Base 3
(76−21, 76, 192)-Net over F3 — Constructive and digital
Digital (55, 76, 192)-net over F3, using
- 31 times duplication [i] based on digital (54, 75, 192)-net over F3, using
- trace code for nets [i] based on digital (4, 25, 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, 25, 64)-net over F27, using
(76−21, 76, 282)-Net over F3 — Digital
Digital (55, 76, 282)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(376, 282, F3, 21) (dual of [282, 206, 22]-code), using
- 34 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 0, 0, 0, 1, 6 times 0, 1, 8 times 0, 1, 11 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]
- 34 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 0, 0, 0, 1, 6 times 0, 1, 8 times 0, 1, 11 times 0) [i] based on linear OA(370, 242, F3, 21) (dual of [242, 172, 22]-code), using
(76−21, 76, 8567)-Net in Base 3 — Upper bound on s
There is no (55, 76, 8568)-net in base 3, because
- 1 times m-reduction [i] would yield (55, 75, 8568)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 608333 813083 923464 098428 067743 494097 > 375 [i]