Best Known (45, 66, s)-Nets in Base 4
(45, 66, 240)-Net over F4 — Constructive and digital
Digital (45, 66, 240)-net over F4, using
- trace code for nets [i] based on digital (1, 22, 80)-net over F64, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 1 and N(F) ≥ 80, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
(45, 66, 288)-Net over F4 — Digital
Digital (45, 66, 288)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(466, 288, F4, 21) (dual of [288, 222, 22]-code), using
- 25 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 1, 4 times 0, 1, 6 times 0, 1, 9 times 0) [i] based on linear OA(459, 256, F4, 21) (dual of [256, 197, 22]-code), using
- an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 255 = 44−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- 25 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 1, 4 times 0, 1, 6 times 0, 1, 9 times 0) [i] based on linear OA(459, 256, F4, 21) (dual of [256, 197, 22]-code), using
(45, 66, 12358)-Net in Base 4 — Upper bound on s
There is no (45, 66, 12359)-net in base 4, because
- 1 times m-reduction [i] would yield (45, 65, 12359)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 1361 550983 683862 262185 404734 300509 136760 > 465 [i]