Best Known (123, 123+33, s)-Nets in Base 4
(123, 123+33, 1052)-Net over F4 — Constructive and digital
Digital (123, 156, 1052)-net over F4, using
- trace code for nets [i] based on digital (6, 39, 263)-net over F256, using
- net from sequence [i] based on digital (6, 262)-sequence over F256, using
(123, 123+33, 4187)-Net over F4 — Digital
Digital (123, 156, 4187)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4156, 4187, F4, 33) (dual of [4187, 4031, 34]-code), using
- 79 step Varšamov–Edel lengthening with (ri) = (4, 1, 1, 0, 0, 1, 4 times 0, 1, 6 times 0, 1, 12 times 0, 1, 18 times 0, 1, 29 times 0) [i] based on linear OA(4145, 4097, F4, 33) (dual of [4097, 3952, 34]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 4097 | 412−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- 79 step Varšamov–Edel lengthening with (ri) = (4, 1, 1, 0, 0, 1, 4 times 0, 1, 6 times 0, 1, 12 times 0, 1, 18 times 0, 1, 29 times 0) [i] based on linear OA(4145, 4097, F4, 33) (dual of [4097, 3952, 34]-code), using
(123, 123+33, 1541239)-Net in Base 4 — Upper bound on s
There is no (123, 156, 1541240)-net in base 4, because
- 1 times m-reduction [i] would yield (123, 155, 1541240)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 2085 944212 454196 240641 464371 299093 576458 369905 610439 399944 639104 713719 519548 235467 188546 106535 > 4155 [i]