Best Known (33−13, 33, s)-Nets in Base 5
(33−13, 33, 104)-Net over F5 — Constructive and digital
Digital (20, 33, 104)-net over F5, using
- 1 times m-reduction [i] based on digital (20, 34, 104)-net over F5, using
- trace code for nets [i] based on digital (3, 17, 52)-net over F25, using
- net from sequence [i] based on digital (3, 51)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 3 and N(F) ≥ 52, using
- net from sequence [i] based on digital (3, 51)-sequence over F25, using
- trace code for nets [i] based on digital (3, 17, 52)-net over F25, using
(33−13, 33, 126)-Net over F5 — Digital
Digital (20, 33, 126)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(533, 126, F5, 13) (dual of [126, 93, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(533, 135, F5, 13) (dual of [135, 102, 14]-code), using
- construction XX applied to C1 = C([122,8]), C2 = C([0,10]), C3 = C1 + C2 = C([0,8]), and C∩ = C1 ∩ C2 = C([122,10]) [i] based on
- linear OA(528, 124, F5, 11) (dual of [124, 96, 12]-code), using the primitive BCH-code C(I) with length 124 = 53−1, defining interval I = {−2,−1,…,8}, and designed minimum distance d ≥ |I|+1 = 12 [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(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(522, 124, F5, 9) (dual of [124, 102, 10]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 124 = 53−1, defining interval I = [0,8], and designed minimum distance d ≥ |I|+1 = 10 [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(51, 4, F5, 1) (dual of [4, 3, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(51, 5, F5, 1) (dual of [5, 4, 2]-code), using
- Reed–Solomon code RS(4,5) [i]
- discarding factors / shortening the dual code based on linear OA(51, 5, F5, 1) (dual of [5, 4, 2]-code), using
- construction XX applied to C1 = C([122,8]), C2 = C([0,10]), C3 = C1 + C2 = C([0,8]), and C∩ = C1 ∩ C2 = C([122,10]) [i] based on
- discarding factors / shortening the dual code based on linear OA(533, 135, F5, 13) (dual of [135, 102, 14]-code), using
(33−13, 33, 3995)-Net in Base 5 — Upper bound on s
There is no (20, 33, 3996)-net in base 5, because
- 1 times m-reduction [i] would yield (20, 32, 3996)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 23301 635206 275243 584065 > 532 [i]