Best Known (48−19, 48, s)-Nets in Base 8
(48−19, 48, 256)-Net over F8 — Constructive and digital
Digital (29, 48, 256)-net over F8, using
- trace code for nets [i] based on digital (5, 24, 128)-net over F64, using
- net from sequence [i] based on digital (5, 127)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 5 and N(F) ≥ 128, using
- net from sequence [i] based on digital (5, 127)-sequence over F64, using
(48−19, 48, 287)-Net over F8 — Digital
Digital (29, 48, 287)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(848, 287, F8, 19) (dual of [287, 239, 20]-code), using
- 27 step Varšamov–Edel lengthening with (ri) = (1, 6 times 0, 1, 19 times 0) [i] based on linear OA(846, 258, F8, 19) (dual of [258, 212, 20]-code), using
- trace code [i] based on linear OA(6423, 129, F64, 19) (dual of [129, 106, 20]-code), using
- extended algebraic-geometric code AGe(F,109P) [i] based on function field F/F64 with g(F) = 4 and N(F) ≥ 129, using
- trace code [i] based on linear OA(6423, 129, F64, 19) (dual of [129, 106, 20]-code), using
- 27 step Varšamov–Edel lengthening with (ri) = (1, 6 times 0, 1, 19 times 0) [i] based on linear OA(846, 258, F8, 19) (dual of [258, 212, 20]-code), using
(48−19, 48, 300)-Net in Base 8 — Constructive
(29, 48, 300)-net in base 8, using
- trace code for nets [i] based on (5, 24, 150)-net in base 64, using
- 4 times m-reduction [i] based on (5, 28, 150)-net in base 64, using
- base change [i] based on digital (1, 24, 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, 24, 150)-net over F128, using
- 4 times m-reduction [i] based on (5, 28, 150)-net in base 64, using
(48−19, 48, 30811)-Net in Base 8 — Upper bound on s
There is no (29, 48, 30812)-net in base 8, because
- 1 times m-reduction [i] would yield (29, 47, 30812)-net in base 8, but
- the generalized Rao bound for nets shows that 8m ≥ 2 787845 827028 041710 954888 808432 055786 019599 > 847 [i]