Best Known (157, 198, s)-Nets in Base 4
(157, 198, 1060)-Net over F4 — Constructive and digital
Digital (157, 198, 1060)-net over F4, using
- 42 times duplication [i] based on digital (155, 196, 1060)-net over F4, using
- trace code for nets [i] based on digital (8, 49, 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, 49, 265)-net over F256, using
(157, 198, 5043)-Net over F4 — Digital
Digital (157, 198, 5043)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4198, 5043, F4, 41) (dual of [5043, 4845, 42]-code), using
- 929 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 1, 0, 0, 1, 5 times 0, 1, 9 times 0, 1, 15 times 0, 1, 23 times 0, 1, 37 times 0, 1, 54 times 0, 1, 76 times 0, 1, 101 times 0, 1, 125 times 0, 1, 143 times 0, 1, 157 times 0, 1, 166 times 0) [i] based on linear OA(4181, 4097, F4, 41) (dual of [4097, 3916, 42]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 4097 | 412−1, defining interval I = [0,20], and minimum distance d ≥ |{−20,−19,…,20}|+1 = 42 (BCH-bound) [i]
- 929 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 1, 0, 0, 1, 5 times 0, 1, 9 times 0, 1, 15 times 0, 1, 23 times 0, 1, 37 times 0, 1, 54 times 0, 1, 76 times 0, 1, 101 times 0, 1, 125 times 0, 1, 143 times 0, 1, 157 times 0, 1, 166 times 0) [i] based on linear OA(4181, 4097, F4, 41) (dual of [4097, 3916, 42]-code), using
(157, 198, 2357615)-Net in Base 4 — Upper bound on s
There is no (157, 198, 2357616)-net in base 4, because
- 1 times m-reduction [i] would yield (157, 197, 2357616)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 40347 947038 099431 234265 746767 935376 726386 580329 539776 969619 983938 301904 606486 781393 520855 452256 562227 476542 127192 921894 > 4197 [i]