Best Known (89, 89+33, s)-Nets in Base 8
(89, 89+33, 1026)-Net over F8 — Constructive and digital
Digital (89, 122, 1026)-net over F8, using
- trace code for nets [i] based on digital (28, 61, 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
(89, 89+33, 5085)-Net over F8 — Digital
Digital (89, 122, 5085)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8122, 5085, F8, 33) (dual of [5085, 4963, 34]-code), using
- 979 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 6 times 0, 1, 18 times 0, 1, 49 times 0, 1, 113 times 0, 1, 201 times 0, 1, 272 times 0, 1, 311 times 0) [i] based on linear OA(8113, 4097, F8, 33) (dual of [4097, 3984, 34]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 4097 | 88−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- 979 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 6 times 0, 1, 18 times 0, 1, 49 times 0, 1, 113 times 0, 1, 201 times 0, 1, 272 times 0, 1, 311 times 0) [i] based on linear OA(8113, 4097, F8, 33) (dual of [4097, 3984, 34]-code), using
(89, 89+33, 6562329)-Net in Base 8 — Upper bound on s
There is no (89, 122, 6562330)-net in base 8, because
- 1 times m-reduction [i] would yield (89, 121, 6562330)-net in base 8, but
- the generalized Rao bound for nets shows that 8m ≥ 18 788342 300343 912463 910164 161679 660324 362680 710180 772297 628783 097970 629817 973461 073557 289655 179795 034437 237247 > 8121 [i]