Best Known (88, 88+33, s)-Nets in Base 8
(88, 88+33, 562)-Net over F8 — Constructive and digital
Digital (88, 121, 562)-net over F8, using
- 1 times m-reduction [i] based on digital (88, 122, 562)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (23, 40, 208)-net over F8, using
- trace code for nets [i] based on digital (3, 20, 104)-net over F64, using
- net from sequence [i] based on digital (3, 103)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 3 and N(F) ≥ 104, using
- net from sequence [i] based on digital (3, 103)-sequence over F64, using
- trace code for nets [i] based on digital (3, 20, 104)-net over F64, using
- digital (48, 82, 354)-net over F8, using
- trace code for nets [i] based on digital (7, 41, 177)-net over F64, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 7 and N(F) ≥ 177, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
- trace code for nets [i] based on digital (7, 41, 177)-net over F64, using
- digital (23, 40, 208)-net over F8, using
- (u, u+v)-construction [i] based on
(88, 88+33, 654)-Net in Base 8 — Constructive
(88, 121, 654)-net in base 8, using
- 81 times duplication [i] based on (87, 120, 654)-net in base 8, using
- (u, u+v)-construction [i] based on
- (24, 40, 300)-net in base 8, using
- trace code for nets [i] based on (4, 20, 150)-net in base 64, using
- 1 times m-reduction [i] based on (4, 21, 150)-net in base 64, using
- base change [i] based on digital (1, 18, 150)-net over F128, using
- net from sequence [i] based on digital (1, 149)-sequence over F128, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 1 and N(F) ≥ 150, using
- net from sequence [i] based on digital (1, 149)-sequence over F128, using
- base change [i] based on digital (1, 18, 150)-net over F128, using
- 1 times m-reduction [i] based on (4, 21, 150)-net in base 64, using
- trace code for nets [i] based on (4, 20, 150)-net in base 64, using
- digital (47, 80, 354)-net over F8, using
- trace code for nets [i] based on digital (7, 40, 177)-net over F64, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 7 and N(F) ≥ 177, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
- trace code for nets [i] based on digital (7, 40, 177)-net over F64, using
- (24, 40, 300)-net in base 8, using
- (u, u+v)-construction [i] based on
(88, 88+33, 4772)-Net over F8 — Digital
Digital (88, 121, 4772)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8121, 4772, F8, 33) (dual of [4772, 4651, 34]-code), using
- 667 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) [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]
- 667 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) [i] based on linear OA(8113, 4097, F8, 33) (dual of [4097, 3984, 34]-code), using
(88, 88+33, 5762551)-Net in Base 8 — Upper bound on s
There is no (88, 121, 5762552)-net in base 8, because
- 1 times m-reduction [i] would yield (88, 120, 5762552)-net in base 8, but
- the generalized Rao bound for nets shows that 8m ≥ 2 348543 262460 947743 052928 806735 013201 535535 241199 792001 247007 034507 688117 225563 246364 762189 112231 142250 166415 > 8120 [i]