Best Known (90, 90+29, s)-Nets in Base 5
(90, 90+29, 408)-Net over F5 — Constructive and digital
Digital (90, 119, 408)-net over F5, using
- 1 times m-reduction [i] based on digital (90, 120, 408)-net over F5, using
- trace code for nets [i] based on digital (30, 60, 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, 60, 204)-net over F25, using
(90, 90+29, 3079)-Net over F5 — Digital
Digital (90, 119, 3079)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5119, 3079, F5, 29) (dual of [3079, 2960, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(5119, 3143, F5, 29) (dual of [3143, 3024, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(25) [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(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(53, 18, F5, 2) (dual of [18, 15, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(53, 24, F5, 2) (dual of [24, 21, 3]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 24 = 52−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- discarding factors / shortening the dual code based on linear OA(53, 24, F5, 2) (dual of [24, 21, 3]-code), using
- construction X applied to Ce(28) ⊂ Ce(25) [i] based on
- discarding factors / shortening the dual code based on linear OA(5119, 3143, F5, 29) (dual of [3143, 3024, 30]-code), using
(90, 90+29, 1176827)-Net in Base 5 — Upper bound on s
There is no (90, 119, 1176828)-net in base 5, because
- 1 times m-reduction [i] would yield (90, 118, 1176828)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 30092 972777 126069 994737 435305 850289 782895 254562 303192 072911 774003 481225 284422 889025 > 5118 [i]