Best Known (73−21, 73, s)-Nets in Base 5
(73−21, 73, 262)-Net over F5 — Constructive and digital
Digital (52, 73, 262)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (1, 11, 10)-net over F5, using
- net from sequence [i] based on digital (1, 9)-sequence over F5, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 1 and N(F) ≥ 10, using
- net from sequence [i] based on digital (1, 9)-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 (1, 11, 10)-net over F5, using
(73−21, 73, 754)-Net over F5 — Digital
Digital (52, 73, 754)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(573, 754, F5, 21) (dual of [754, 681, 22]-code), using
- 117 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 4 times 0, 1, 9 times 0, 1, 19 times 0, 1, 32 times 0, 1, 46 times 0) [i] based on linear OA(566, 630, F5, 21) (dual of [630, 564, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(18) [i] based on
- linear OA(565, 625, F5, 21) (dual of [625, 560, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(561, 625, F5, 19) (dual of [625, 564, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(51, 5, F5, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(51, s, F5, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(20) ⊂ Ce(18) [i] based on
- 117 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 4 times 0, 1, 9 times 0, 1, 19 times 0, 1, 32 times 0, 1, 46 times 0) [i] based on linear OA(566, 630, F5, 21) (dual of [630, 564, 22]-code), using
(73−21, 73, 122032)-Net in Base 5 — Upper bound on s
There is no (52, 73, 122033)-net in base 5, because
- 1 times m-reduction [i] would yield (52, 72, 122033)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 211 767283 099090 423307 040061 049374 682592 867851 683065 > 572 [i]