Best Known (54, 54+16, s)-Nets in Base 4
(54, 54+16, 1032)-Net over F4 — Constructive and digital
Digital (54, 70, 1032)-net over F4, using
- 42 times duplication [i] based on digital (52, 68, 1032)-net over F4, using
- trace code for nets [i] based on digital (1, 17, 258)-net over F256, using
- net from sequence [i] based on digital (1, 257)-sequence over F256, using
- trace code for nets [i] based on digital (1, 17, 258)-net over F256, using
(54, 54+16, 1395)-Net over F4 — Digital
Digital (54, 70, 1395)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(470, 1395, F4, 16) (dual of [1395, 1325, 17]-code), using
- 361 step Varšamov–Edel lengthening with (ri) = (2, 1, 0, 0, 1, 6 times 0, 1, 12 times 0, 1, 24 times 0, 1, 42 times 0, 1, 65 times 0, 1, 90 times 0, 1, 111 times 0) [i] based on linear OA(460, 1024, F4, 16) (dual of [1024, 964, 17]-code), using
- 1 times truncation [i] based on linear OA(461, 1025, F4, 17) (dual of [1025, 964, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 1025 | 410−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- 1 times truncation [i] based on linear OA(461, 1025, F4, 17) (dual of [1025, 964, 18]-code), using
- 361 step Varšamov–Edel lengthening with (ri) = (2, 1, 0, 0, 1, 6 times 0, 1, 12 times 0, 1, 24 times 0, 1, 42 times 0, 1, 65 times 0, 1, 90 times 0, 1, 111 times 0) [i] based on linear OA(460, 1024, F4, 16) (dual of [1024, 964, 17]-code), using
(54, 54+16, 232585)-Net in Base 4 — Upper bound on s
There is no (54, 70, 232586)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 1 393831 219273 712253 440178 104496 631951 533145 > 470 [i]