Best Known (53−21, 53, s)-Nets in Base 8
(53−21, 53, 256)-Net over F8 — Constructive and digital
Digital (32, 53, 256)-net over F8, using
- 1 times m-reduction [i] based on digital (32, 54, 256)-net over F8, using
- trace code for nets [i] based on digital (5, 27, 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
- trace code for nets [i] based on digital (5, 27, 128)-net over F64, using
(53−21, 53, 300)-Net in Base 8 — Constructive
(32, 53, 300)-net in base 8, using
- 1 times m-reduction [i] based on (32, 54, 300)-net in base 8, using
- trace code for nets [i] based on (5, 27, 150)-net in base 64, using
- 1 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
- 1 times m-reduction [i] based on (5, 28, 150)-net in base 64, using
- trace code for nets [i] based on (5, 27, 150)-net in base 64, using
(53−21, 53, 304)-Net over F8 — Digital
Digital (32, 53, 304)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(853, 304, F8, 21) (dual of [304, 251, 22]-code), using
- 43 step Varšamov–Edel lengthening with (ri) = (1, 0, 0, 1, 14 times 0, 1, 24 times 0) [i] based on linear OA(850, 258, F8, 21) (dual of [258, 208, 22]-code), using
- trace code [i] based on linear OA(6425, 129, F64, 21) (dual of [129, 104, 22]-code), using
- extended algebraic-geometric code AGe(F,107P) [i] based on function field F/F64 with g(F) = 4 and N(F) ≥ 129, using
- trace code [i] based on linear OA(6425, 129, F64, 21) (dual of [129, 104, 22]-code), using
- 43 step Varšamov–Edel lengthening with (ri) = (1, 0, 0, 1, 14 times 0, 1, 24 times 0) [i] based on linear OA(850, 258, F8, 21) (dual of [258, 208, 22]-code), using
(53−21, 53, 32126)-Net in Base 8 — Upper bound on s
There is no (32, 53, 32127)-net in base 8, because
- 1 times m-reduction [i] would yield (32, 52, 32127)-net in base 8, but
- the generalized Rao bound for nets shows that 8m ≥ 91352 493348 876861 523207 824764 481673 428967 715659 > 852 [i]