Best Known (98−22, 98, s)-Nets in Base 5
(98−22, 98, 400)-Net over F5 — Constructive and digital
Digital (76, 98, 400)-net over F5, using
- 4 times m-reduction [i] based on digital (76, 102, 400)-net over F5, using
- trace code for nets [i] based on digital (25, 51, 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, 51, 200)-net over F25, using
(98−22, 98, 3998)-Net over F5 — Digital
Digital (76, 98, 3998)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(598, 3998, F5, 22) (dual of [3998, 3900, 23]-code), using
- 856 step Varšamov–Edel lengthening with (ri) = (3, 0, 1, 0, 0, 0, 1, 8 times 0, 1, 18 times 0, 1, 35 times 0, 1, 62 times 0, 1, 102 times 0, 1, 152 times 0, 1, 207 times 0, 1, 258 times 0) [i] based on linear OA(586, 3130, F5, 22) (dual of [3130, 3044, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(20) [i] based on
- linear OA(586, 3125, F5, 22) (dual of [3125, 3039, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(581, 3125, F5, 21) (dual of [3125, 3044, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [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(21) ⊂ Ce(20) [i] based on
- 856 step Varšamov–Edel lengthening with (ri) = (3, 0, 1, 0, 0, 0, 1, 8 times 0, 1, 18 times 0, 1, 35 times 0, 1, 62 times 0, 1, 102 times 0, 1, 152 times 0, 1, 207 times 0, 1, 258 times 0) [i] based on linear OA(586, 3130, F5, 22) (dual of [3130, 3044, 23]-code), using
(98−22, 98, 2070808)-Net in Base 5 — Upper bound on s
There is no (76, 98, 2070809)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 315 545904 116540 743131 408474 630743 381997 813382 422451 286306 239741 975485 > 598 [i]