Best Known (59−18, 59, s)-Nets in Base 5
(59−18, 59, 252)-Net over F5 — Constructive and digital
Digital (41, 59, 252)-net over F5, using
- 3 times m-reduction [i] based on digital (41, 62, 252)-net over F5, using
- trace code for nets [i] based on digital (10, 31, 126)-net over F25, using
- net from sequence [i] based on digital (10, 125)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 10 and N(F) ≥ 126, using
- the Hermitian function field over F25 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 10 and N(F) ≥ 126, using
- net from sequence [i] based on digital (10, 125)-sequence over F25, using
- trace code for nets [i] based on digital (10, 31, 126)-net over F25, using
(59−18, 59, 571)-Net over F5 — Digital
Digital (41, 59, 571)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(559, 571, F5, 18) (dual of [571, 512, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(559, 638, F5, 18) (dual of [638, 579, 19]-code), using
- construction XX applied to C1 = C([622,13]), C2 = C([0,15]), C3 = C1 + C2 = C([0,13]), and C∩ = C1 ∩ C2 = C([622,15]) [i] based on
- linear OA(553, 624, F5, 16) (dual of [624, 571, 17]-code), using the primitive BCH-code C(I) with length 624 = 54−1, defining interval I = {−2,−1,…,13}, and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(549, 624, F5, 16) (dual of [624, 575, 17]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [0,15], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(557, 624, F5, 18) (dual of [624, 567, 19]-code), using the primitive BCH-code C(I) with length 624 = 54−1, defining interval I = {−2,−1,…,15}, and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(545, 624, F5, 14) (dual of [624, 579, 15]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [0,13], and designed minimum distance d ≥ |I|+1 = 15 [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(51, 5, F5, 1) (dual of [5, 4, 2]-code), using
- Reed–Solomon code RS(4,5) [i]
- construction XX applied to C1 = C([622,13]), C2 = C([0,15]), C3 = C1 + C2 = C([0,13]), and C∩ = C1 ∩ C2 = C([622,15]) [i] based on
- discarding factors / shortening the dual code based on linear OA(559, 638, F5, 18) (dual of [638, 579, 19]-code), using
(59−18, 59, 39605)-Net in Base 5 — Upper bound on s
There is no (41, 59, 39606)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 173477 220210 696410 507548 937291 598013 752025 > 559 [i]