Best Known (57, 57+21, s)-Nets in Base 5
(57, 57+21, 273)-Net over F5 — Constructive and digital
Digital (57, 78, 273)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (6, 16, 21)-net over F5, using
- net from sequence [i] based on digital (6, 20)-sequence over F5, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 6 and N(F) ≥ 21, using
- net from sequence [i] based on digital (6, 20)-sequence over F5, using
- digital (41, 62, 252)-net over F5, using
- trace code for nets [i] based on digital (10, 31, 126)-net over F25, using
- net from sequence [i] based on digital (10, 125)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 10 and N(F) ≥ 126, using
- the Hermitian function field over F25 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 10 and N(F) ≥ 126, using
- net from sequence [i] based on digital (10, 125)-sequence over F25, using
- trace code for nets [i] based on digital (10, 31, 126)-net over F25, using
- digital (6, 16, 21)-net over F5, using
(57, 57+21, 1115)-Net over F5 — Digital
Digital (57, 78, 1115)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(578, 1115, F5, 21) (dual of [1115, 1037, 22]-code), using
- 1036 step Varšamov–Edel lengthening with (ri) = (5, 2, 2, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 4 times 0, 1, 4 times 0, 1, 5 times 0, 1, 6 times 0, 1, 6 times 0, 1, 7 times 0, 1, 8 times 0, 1, 8 times 0, 1, 10 times 0, 1, 10 times 0, 1, 11 times 0, 1, 13 times 0, 1, 14 times 0, 1, 15 times 0, 1, 16 times 0, 1, 18 times 0, 1, 20 times 0, 1, 22 times 0, 1, 23 times 0, 1, 26 times 0, 1, 28 times 0, 1, 31 times 0, 1, 33 times 0, 1, 36 times 0, 1, 39 times 0, 1, 43 times 0, 1, 47 times 0, 1, 51 times 0, 1, 55 times 0, 1, 60 times 0, 1, 65 times 0, 1, 70 times 0, 1, 77 times 0, 1, 84 times 0) [i] based on linear OA(521, 22, F5, 21) (dual of [22, 1, 22]-code or 22-arc in PG(20,5)), using
- dual of repetition code with length 22 [i]
- 1036 step Varšamov–Edel lengthening with (ri) = (5, 2, 2, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 4 times 0, 1, 4 times 0, 1, 5 times 0, 1, 6 times 0, 1, 6 times 0, 1, 7 times 0, 1, 8 times 0, 1, 8 times 0, 1, 10 times 0, 1, 10 times 0, 1, 11 times 0, 1, 13 times 0, 1, 14 times 0, 1, 15 times 0, 1, 16 times 0, 1, 18 times 0, 1, 20 times 0, 1, 22 times 0, 1, 23 times 0, 1, 26 times 0, 1, 28 times 0, 1, 31 times 0, 1, 33 times 0, 1, 36 times 0, 1, 39 times 0, 1, 43 times 0, 1, 47 times 0, 1, 51 times 0, 1, 55 times 0, 1, 60 times 0, 1, 65 times 0, 1, 70 times 0, 1, 77 times 0, 1, 84 times 0) [i] based on linear OA(521, 22, F5, 21) (dual of [22, 1, 22]-code or 22-arc in PG(20,5)), using
(57, 57+21, 272881)-Net in Base 5 — Upper bound on s
There is no (57, 78, 272882)-net in base 5, because
- 1 times m-reduction [i] would yield (57, 77, 272882)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 661754 918585 986302 534500 940460 432553 021183 622281 484225 > 577 [i]