Best Known (37−14, 37, s)-Nets in Base 5
(37−14, 37, 132)-Net over F5 — Constructive and digital
Digital (23, 37, 132)-net over F5, using
- 1 times m-reduction [i] based on digital (23, 38, 132)-net over F5, using
- trace code for nets [i] based on digital (4, 19, 66)-net over F25, using
- net from sequence [i] based on digital (4, 65)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 4 and N(F) ≥ 66, using
- net from sequence [i] based on digital (4, 65)-sequence over F25, using
- trace code for nets [i] based on digital (4, 19, 66)-net over F25, using
(37−14, 37, 148)-Net over F5 — Digital
Digital (23, 37, 148)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(537, 148, F5, 14) (dual of [148, 111, 15]-code), using
- 12 step Varšamov–Edel lengthening with (ri) = (1, 0, 0, 1, 8 times 0) [i] based on linear OA(535, 134, F5, 14) (dual of [134, 99, 15]-code), using
- construction XX applied to C1 = C([122,10]), C2 = C([0,11]), C3 = C1 + C2 = C([0,10]), and C∩ = C1 ∩ C2 = C([122,11]) [i] based on
- linear OA(531, 124, F5, 13) (dual of [124, 93, 14]-code), using the primitive BCH-code C(I) with length 124 = 53−1, defining interval I = {−2,−1,…,10}, and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(528, 124, F5, 12) (dual of [124, 96, 13]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 124 = 53−1, defining interval I = [0,11], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(534, 124, F5, 14) (dual of [124, 90, 15]-code), using the primitive BCH-code C(I) with length 124 = 53−1, defining interval I = {−2,−1,…,11}, and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(525, 124, F5, 11) (dual of [124, 99, 12]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 124 = 53−1, defining interval I = [0,10], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(51, 7, F5, 1) (dual of [7, 6, 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, 3, F5, 0) (dual of [3, 3, 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([122,10]), C2 = C([0,11]), C3 = C1 + C2 = C([0,10]), and C∩ = C1 ∩ C2 = C([122,11]) [i] based on
- 12 step Varšamov–Edel lengthening with (ri) = (1, 0, 0, 1, 8 times 0) [i] based on linear OA(535, 134, F5, 14) (dual of [134, 99, 15]-code), using
(37−14, 37, 4177)-Net in Base 5 — Upper bound on s
There is no (23, 37, 4178)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 72 815069 917809 542761 954665 > 537 [i]