Best Known (122−29, 122, s)-Nets in Base 5
(122−29, 122, 416)-Net over F5 — Constructive and digital
Digital (93, 122, 416)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (32, 46, 208)-net over F5, using
- trace code for nets [i] based on digital (9, 23, 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, 23, 104)-net over F25, using
- digital (47, 76, 208)-net over F5, using
- trace code for nets [i] based on digital (9, 38, 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, 38, 104)-net over F25, using
- digital (32, 46, 208)-net over F5, using
(122−29, 122, 3270)-Net over F5 — Digital
Digital (93, 122, 3270)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5122, 3270, F5, 29) (dual of [3270, 3148, 30]-code), using
- 134 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 6 times 0, 1, 16 times 0, 1, 36 times 0, 1, 69 times 0) [i] based on linear OA(5116, 3130, F5, 29) (dual of [3130, 3014, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(27) [i] based on
- linear OA(5116, 3125, F5, 29) (dual of [3125, 3009, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(5111, 3125, F5, 28) (dual of [3125, 3014, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [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(28) ⊂ Ce(27) [i] based on
- 134 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 6 times 0, 1, 16 times 0, 1, 36 times 0, 1, 69 times 0) [i] based on linear OA(5116, 3130, F5, 29) (dual of [3130, 3014, 30]-code), using
(122−29, 122, 1661472)-Net in Base 5 — Upper bound on s
There is no (93, 122, 1661473)-net in base 5, because
- 1 times m-reduction [i] would yield (93, 121, 1661473)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 3 761613 363244 896259 829934 329815 796662 223620 538915 605014 697006 565725 910396 075874 451945 > 5121 [i]