Best Known (121−31, 121, s)-Nets in Base 5
(121−31, 121, 400)-Net over F5 — Constructive and digital
Digital (90, 121, 400)-net over F5, using
- 9 times m-reduction [i] based on digital (90, 130, 400)-net over F5, using
- trace code for nets [i] based on digital (25, 65, 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, 65, 200)-net over F25, using
(121−31, 121, 2257)-Net over F5 — Digital
Digital (90, 121, 2257)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5121, 2257, F5, 31) (dual of [2257, 2136, 32]-code), using
- discarding factors / shortening the dual code 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]
- discarding factors / shortening the dual code based on linear OA(5121, 3125, F5, 31) (dual of [3125, 3004, 32]-code), using
(121−31, 121, 627276)-Net in Base 5 — Upper bound on s
There is no (90, 121, 627277)-net in base 5, because
- 1 times m-reduction [i] would yield (90, 120, 627277)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 752319 563539 393613 976767 129839 348828 655856 089253 503750 141386 183570 330043 361254 237565 > 5120 [i]