Best Known (122, 139, s)-Nets in Base 5
(122, 139, 1048581)-Net over F5 — Constructive and digital
Digital (122, 139, 1048581)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (0, 8, 6)-net over F5, using
- net from sequence [i] based on digital (0, 5)-sequence over F5, using
- Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 0 and N(F) ≥ 6, using
- the rational function field F5(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 5)-sequence over F5, using
- digital (114, 131, 1048575)-net over F5, using
- net defined by OOA [i] based on linear OOA(5131, 1048575, F5, 17, 17) (dual of [(1048575, 17), 17825644, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(5131, 8388601, F5, 17) (dual of [8388601, 8388470, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(5131, large, F5, 17) (dual of [large, large−131, 18]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 9765624 = 510−1, defining interval I = [0,16], and designed minimum distance d ≥ |I|+1 = 18 [i]
- discarding factors / shortening the dual code based on linear OA(5131, large, F5, 17) (dual of [large, large−131, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(5131, 8388601, F5, 17) (dual of [8388601, 8388470, 18]-code), using
- net defined by OOA [i] based on linear OOA(5131, 1048575, F5, 17, 17) (dual of [(1048575, 17), 17825644, 18]-NRT-code), using
- digital (0, 8, 6)-net over F5, using
(122, 139, 4327426)-Net over F5 — Digital
Digital (122, 139, 4327426)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5139, 4327426, F5, 17) (dual of [4327426, 4327287, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(5139, large, F5, 17) (dual of [large, large−139, 18]-code), using
- 8 times code embedding in larger space [i] based on linear OA(5131, large, F5, 17) (dual of [large, large−131, 18]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 9765624 = 510−1, defining interval I = [0,16], and designed minimum distance d ≥ |I|+1 = 18 [i]
- 8 times code embedding in larger space [i] based on linear OA(5131, large, F5, 17) (dual of [large, large−131, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(5139, large, F5, 17) (dual of [large, large−139, 18]-code), using
(122, 139, large)-Net in Base 5 — Upper bound on s
There is no (122, 139, large)-net in base 5, because
- 15 times m-reduction [i] would yield (122, 124, large)-net in base 5, but