Best Known (134−33, 134, s)-Nets in Base 5
(134−33, 134, 416)-Net over F5 — Constructive and digital
Digital (101, 134, 416)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (34, 50, 208)-net over F5, using
- trace code for nets [i] based on digital (9, 25, 104)-net over F25, using
- net from sequence [i] based on digital (9, 103)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 9 and N(F) ≥ 104, using
- net from sequence [i] based on digital (9, 103)-sequence over F25, using
- trace code for nets [i] based on digital (9, 25, 104)-net over F25, using
- digital (51, 84, 208)-net over F5, using
- trace code for nets [i] based on digital (9, 42, 104)-net over F25, using
- net from sequence [i] based on digital (9, 103)-sequence over F25 (see above)
- trace code for nets [i] based on digital (9, 42, 104)-net over F25, using
- digital (34, 50, 208)-net over F5, using
(134−33, 134, 3074)-Net over F5 — Digital
Digital (101, 134, 3074)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5134, 3074, F5, 33) (dual of [3074, 2940, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(5134, 3139, F5, 33) (dual of [3139, 3005, 34]-code), using
- 2 times code embedding in larger space [i] based on linear OA(5132, 3137, F5, 33) (dual of [3137, 3005, 34]-code), using
- construction X applied to C([0,16]) ⊂ C([0,15]) [i] based on
- linear OA(5131, 3126, F5, 33) (dual of [3126, 2995, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 3126 | 510−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- linear OA(5121, 3126, F5, 31) (dual of [3126, 3005, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 3126 | 510−1, defining interval I = [0,15], and minimum distance d ≥ |{−15,−14,…,15}|+1 = 32 (BCH-bound) [i]
- linear OA(51, 11, F5, 1) (dual of [11, 10, 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 C([0,16]) ⊂ C([0,15]) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(5132, 3137, F5, 33) (dual of [3137, 3005, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(5134, 3139, F5, 33) (dual of [3139, 3005, 34]-code), using
(134−33, 134, 1098151)-Net in Base 5 — Upper bound on s
There is no (101, 134, 1098152)-net in base 5, because
- 1 times m-reduction [i] would yield (101, 133, 1098152)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 918 358699 826466 763136 331850 518217 322791 011767 387960 428018 207044 608702 155979 314978 429489 424385 > 5133 [i]