Best Known (34−10, 34, s)-Nets in Base 5
(34−10, 34, 144)-Net over F5 — Constructive and digital
Digital (24, 34, 144)-net over F5, using
- generalized (u, u+v)-construction [i] based on
- digital (2, 5, 46)-net over F5, using
- s-reduction based on digital (2, 5, 66)-net over F5, using
- net defined by OOA [i] based on linear OOA(55, 66, F5, 3, 3) (dual of [(66, 3), 193, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(55, 66, F5, 2, 3) (dual of [(66, 2), 127, 4]-NRT-code), using
- net defined by OOA [i] based on linear OOA(55, 66, F5, 3, 3) (dual of [(66, 3), 193, 4]-NRT-code), using
- s-reduction based on digital (2, 5, 66)-net over F5, using
- digital (4, 9, 46)-net over F5, using
- digital (10, 20, 52)-net over F5, using
- trace code for nets [i] based on digital (0, 10, 26)-net over F25, using
- net from sequence [i] based on digital (0, 25)-sequence over F25, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 0 and N(F) ≥ 26, using
- the rational function field F25(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 25)-sequence over F25, using
- trace code for nets [i] based on digital (0, 10, 26)-net over F25, using
- digital (2, 5, 46)-net over F5, using
(34−10, 34, 637)-Net over F5 — Digital
Digital (24, 34, 637)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(534, 637, F5, 10) (dual of [637, 603, 11]-code), using
- construction XX applied to C1 = C([622,6]), C2 = C([0,7]), C3 = C1 + C2 = C([0,6]), and C∩ = C1 ∩ C2 = C([622,7]) [i] based on
- linear OA(529, 624, F5, 9) (dual of [624, 595, 10]-code), using the primitive BCH-code C(I) with length 624 = 54−1, defining interval I = {−2,−1,…,6}, and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(525, 624, F5, 8) (dual of [624, 599, 9]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [0,7], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(533, 624, F5, 10) (dual of [624, 591, 11]-code), using the primitive BCH-code C(I) with length 624 = 54−1, defining interval I = {−2,−1,…,7}, and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(521, 624, F5, 7) (dual of [624, 603, 8]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [0,6], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(51, 9, F5, 1) (dual of [9, 8, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(51, s, F5, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- linear OA(50, 4, F5, 0) (dual of [4, 4, 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 XX applied to C1 = C([622,6]), C2 = C([0,7]), C3 = C1 + C2 = C([0,6]), and C∩ = C1 ∩ C2 = C([622,7]) [i] based on
(34−10, 34, 36875)-Net in Base 5 — Upper bound on s
There is no (24, 34, 36876)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 582142 230960 912101 992625 > 534 [i]