Best Known (180, 180+49, s)-Nets in Base 4
(180, 180+49, 1060)-Net over F4 — Constructive and digital
Digital (180, 229, 1060)-net over F4, using
- 41 times duplication [i] based on digital (179, 228, 1060)-net over F4, using
- trace code for nets [i] based on digital (8, 57, 265)-net over F256, using
- net from sequence [i] based on digital (8, 264)-sequence over F256, using
- trace code for nets [i] based on digital (8, 57, 265)-net over F256, using
(180, 180+49, 4681)-Net over F4 — Digital
Digital (180, 229, 4681)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4229, 4681, F4, 49) (dual of [4681, 4452, 50]-code), using
- 572 step Varšamov–Edel lengthening with (ri) = (2, 1, 0, 1, 4 times 0, 1, 9 times 0, 1, 19 times 0, 1, 34 times 0, 1, 58 times 0, 1, 83 times 0, 1, 105 times 0, 1, 120 times 0, 1, 128 times 0) [i] based on linear OA(4217, 4097, F4, 49) (dual of [4097, 3880, 50]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 4097 | 412−1, defining interval I = [0,24], and minimum distance d ≥ |{−24,−23,…,24}|+1 = 50 (BCH-bound) [i]
- 572 step Varšamov–Edel lengthening with (ri) = (2, 1, 0, 1, 4 times 0, 1, 9 times 0, 1, 19 times 0, 1, 34 times 0, 1, 58 times 0, 1, 83 times 0, 1, 105 times 0, 1, 120 times 0, 1, 128 times 0) [i] based on linear OA(4217, 4097, F4, 49) (dual of [4097, 3880, 50]-code), using
(180, 180+49, 1713193)-Net in Base 4 — Upper bound on s
There is no (180, 229, 1713194)-net in base 4, because
- 1 times m-reduction [i] would yield (180, 228, 1713194)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 186071 626243 810468 917428 160762 318489 795008 871697 676342 997429 389492 659376 786203 520188 241105 576124 431895 813611 059020 463561 940942 598538 841026 > 4228 [i]