Best Known (126−31, 126, s)-Nets in Base 5
(126−31, 126, 408)-Net over F5 — Constructive and digital
Digital (95, 126, 408)-net over F5, using
- 4 times m-reduction [i] based on digital (95, 130, 408)-net over F5, using
- trace code for nets [i] based on digital (30, 65, 204)-net over F25, using
- net from sequence [i] based on digital (30, 203)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 30 and N(F) ≥ 204, using
- net from sequence [i] based on digital (30, 203)-sequence over F25, using
- trace code for nets [i] based on digital (30, 65, 204)-net over F25, using
(126−31, 126, 2985)-Net over F5 — Digital
Digital (95, 126, 2985)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5126, 2985, F5, 31) (dual of [2985, 2859, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(5126, 3146, F5, 31) (dual of [3146, 3020, 32]-code), using
- construction XX applied to Ce(30) ⊂ Ce(26) ⊂ Ce(25) [i] based on
- linear OA(5121, 3125, F5, 31) (dual of [3125, 3004, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(5106, 3125, F5, 27) (dual of [3125, 3019, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(5101, 3125, F5, 26) (dual of [3125, 3024, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(54, 20, F5, 3) (dual of [20, 16, 4]-code or 20-cap in PG(3,5)), using
- linear OA(50, 1, F5, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(30) ⊂ Ce(26) ⊂ Ce(25) [i] based on
- discarding factors / shortening the dual code based on linear OA(5126, 3146, F5, 31) (dual of [3146, 3020, 32]-code), using
(126−31, 126, 1072635)-Net in Base 5 — Upper bound on s
There is no (95, 126, 1072636)-net in base 5, because
- 1 times m-reduction [i] would yield (95, 125, 1072636)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 2350 991598 199216 573963 900805 084619 876150 580635 981544 816965 021472 764985 310067 039928 691825 > 5125 [i]