Best Known (121, 121+49, s)-Nets in Base 8
(121, 121+49, 1026)-Net over F8 — Constructive and digital
Digital (121, 170, 1026)-net over F8, using
- t-expansion [i] based on digital (114, 170, 1026)-net over F8, using
- 2 times m-reduction [i] based on digital (114, 172, 1026)-net over F8, using
- trace code for nets [i] based on digital (28, 86, 513)-net over F64, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- the Hermitian function field over F64 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- trace code for nets [i] based on digital (28, 86, 513)-net over F64, using
- 2 times m-reduction [i] based on digital (114, 172, 1026)-net over F8, using
(121, 121+49, 4256)-Net over F8 — Digital
Digital (121, 170, 4256)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8170, 4256, F8, 49) (dual of [4256, 4086, 50]-code), using
- 158 step Varšamov–Edel lengthening with (ri) = (1, 157 times 0) [i] based on linear OA(8169, 4097, F8, 49) (dual of [4097, 3928, 50]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 4097 | 88−1, defining interval I = [0,24], and minimum distance d ≥ |{−24,−23,…,24}|+1 = 50 (BCH-bound) [i]
- 158 step Varšamov–Edel lengthening with (ri) = (1, 157 times 0) [i] based on linear OA(8169, 4097, F8, 49) (dual of [4097, 3928, 50]-code), using
(121, 121+49, 3202737)-Net in Base 8 — Upper bound on s
There is no (121, 170, 3202738)-net in base 8, because
- 1 times m-reduction [i] would yield (121, 169, 3202738)-net in base 8, but
- the generalized Rao bound for nets shows that 8m ≥ 418 995904 164868 082224 562245 370100 658722 578888 204004 932397 116357 971043 253591 592531 505407 947175 894490 546665 050495 730242 768756 754107 415460 826631 803036 524236 > 8169 [i]