Best Known (54, 54+33, s)-Nets in Base 8
(54, 54+33, 354)-Net over F8 — Constructive and digital
Digital (54, 87, 354)-net over F8, using
- 7 times m-reduction [i] based on digital (54, 94, 354)-net over F8, using
- trace code for nets [i] based on digital (7, 47, 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, 47, 177)-net over F64, using
(54, 54+33, 384)-Net in Base 8 — Constructive
(54, 87, 384)-net in base 8, using
- 1 times m-reduction [i] based on (54, 88, 384)-net in base 8, using
- trace code for nets [i] based on (10, 44, 192)-net in base 64, using
- 5 times m-reduction [i] based on (10, 49, 192)-net in base 64, using
- base change [i] based on digital (3, 42, 192)-net over F128, using
- net from sequence [i] based on digital (3, 191)-sequence over F128, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 3 and N(F) ≥ 192, using
- net from sequence [i] based on digital (3, 191)-sequence over F128, using
- base change [i] based on digital (3, 42, 192)-net over F128, using
- 5 times m-reduction [i] based on (10, 49, 192)-net in base 64, using
- trace code for nets [i] based on (10, 44, 192)-net in base 64, using
(54, 54+33, 541)-Net over F8 — Digital
Digital (54, 87, 541)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(887, 541, F8, 33) (dual of [541, 454, 34]-code), using
- 26 step Varšamov–Edel lengthening with (ri) = (1, 5 times 0, 1, 19 times 0) [i] based on linear OA(885, 513, F8, 33) (dual of [513, 428, 34]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 513 | 86−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- 26 step Varšamov–Edel lengthening with (ri) = (1, 5 times 0, 1, 19 times 0) [i] based on linear OA(885, 513, F8, 33) (dual of [513, 428, 34]-code), using
(54, 54+33, 69420)-Net in Base 8 — Upper bound on s
There is no (54, 87, 69421)-net in base 8, because
- 1 times m-reduction [i] would yield (54, 86, 69421)-net in base 8, but
- the generalized Rao bound for nets shows that 8m ≥ 463214 646092 996821 481778 718386 522952 566931 609925 568635 580819 548793 672978 210698 > 886 [i]