Best Known (135−33, 135, s)-Nets in Base 5
(135−33, 135, 416)-Net over F5 — Constructive and digital
Digital (102, 135, 416)-net over F5, using
- 51 times duplication [i] based on 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
- (u, u+v)-construction [i] based on
(135−33, 135, 3159)-Net over F5 — Digital
Digital (102, 135, 3159)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5135, 3159, F5, 33) (dual of [3159, 3024, 34]-code), using
- 25 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 6 times 0, 1, 14 times 0) [i] based on linear OA(5131, 3130, F5, 33) (dual of [3130, 2999, 34]-code), using
- construction X applied to Ce(32) ⊂ Ce(31) [i] based on
- linear OA(5131, 3125, F5, 33) (dual of [3125, 2994, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(5126, 3125, F5, 32) (dual of [3125, 2999, 33]-code), using an extension Ce(31) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,31], and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(50, 5, F5, 0) (dual of [5, 5, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(50, s, F5, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(32) ⊂ Ce(31) [i] based on
- 25 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 6 times 0, 1, 14 times 0) [i] based on linear OA(5131, 3130, F5, 33) (dual of [3130, 2999, 34]-code), using
(135−33, 135, 1214362)-Net in Base 5 — Upper bound on s
There is no (102, 135, 1214363)-net in base 5, because
- 1 times m-reduction [i] would yield (102, 134, 1214363)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 4591 795026 516701 124825 910724 700265 485570 866323 285877 918532 036354 720094 156419 949313 835429 907905 > 5134 [i]