Best Known (124, 124+33, s)-Nets in Base 4
(124, 124+33, 1052)-Net over F4 — Constructive and digital
Digital (124, 157, 1052)-net over F4, using
- 41 times duplication [i] based on 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
- trace code for nets [i] based on digital (6, 39, 263)-net over F256, using
(124, 124+33, 4232)-Net over F4 — Digital
Digital (124, 157, 4232)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4157, 4232, F4, 33) (dual of [4232, 4075, 34]-code), using
- 123 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, 1, 43 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]
- 123 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, 1, 43 times 0) [i] based on linear OA(4145, 4097, F4, 33) (dual of [4097, 3952, 34]-code), using
(124, 124+33, 1680734)-Net in Base 4 — Upper bound on s
There is no (124, 157, 1680735)-net in base 4, because
- 1 times m-reduction [i] would yield (124, 156, 1680735)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 8343 753068 619609 088732 126485 985458 547076 917165 681604 549107 900526 592139 549018 994693 911331 030743 > 4156 [i]