Best Known (82, 106, s)-Nets in Base 5
(82, 106, 400)-Net over F5 — Constructive and digital
Digital (82, 106, 400)-net over F5, using
- 8 times m-reduction [i] based on digital (82, 114, 400)-net over F5, using
- trace code for nets [i] based on digital (25, 57, 200)-net over F25, using
- net from sequence [i] based on digital (25, 199)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 25 and N(F) ≥ 200, using
- net from sequence [i] based on digital (25, 199)-sequence over F25, using
- trace code for nets [i] based on digital (25, 57, 200)-net over F25, using
(82, 106, 3941)-Net over F5 — Digital
Digital (82, 106, 3941)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5106, 3941, F5, 24) (dual of [3941, 3835, 25]-code), using
- 801 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 4 times 0, 1, 10 times 0, 1, 23 times 0, 1, 49 times 0, 1, 94 times 0, 1, 153 times 0, 1, 209 times 0, 1, 249 times 0) [i] based on linear OA(596, 3130, F5, 24) (dual of [3130, 3034, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(22) [i] based on
- linear OA(596, 3125, F5, 24) (dual of [3125, 3029, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(591, 3125, F5, 23) (dual of [3125, 3034, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [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(23) ⊂ Ce(22) [i] based on
- 801 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 4 times 0, 1, 10 times 0, 1, 23 times 0, 1, 49 times 0, 1, 94 times 0, 1, 153 times 0, 1, 209 times 0, 1, 249 times 0) [i] based on linear OA(596, 3130, F5, 24) (dual of [3130, 3034, 25]-code), using
(82, 106, 1974852)-Net in Base 5 — Upper bound on s
There is no (82, 106, 1974853)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 123 260138 532335 356286 448070 097295 891490 893530 431151 848503 124116 493772 211665 > 5106 [i]