Best Known (48, 70, s)-Nets in Base 5
(48, 70, 252)-Net over F5 — Constructive and digital
Digital (48, 70, 252)-net over F5, using
- 6 times m-reduction [i] based on digital (48, 76, 252)-net over F5, using
- trace code for nets [i] based on digital (10, 38, 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, 38, 126)-net over F25, using
(48, 70, 522)-Net over F5 — Digital
Digital (48, 70, 522)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(570, 522, F5, 22) (dual of [522, 452, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(570, 633, F5, 22) (dual of [633, 563, 23]-code), using
- construction XX applied to C1 = C([136,156]), C2 = C([138,157]), C3 = C1 + C2 = C([138,156]), and C∩ = C1 ∩ C2 = C([136,157]) [i] based on
- linear OA(565, 624, F5, 21) (dual of [624, 559, 22]-code), using the primitive BCH-code C(I) with length 624 = 54−1, defining interval I = {136,137,…,156}, and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(565, 624, F5, 20) (dual of [624, 559, 21]-code), using the primitive BCH-code C(I) with length 624 = 54−1, defining interval I = {138,139,…,157}, and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(569, 624, F5, 22) (dual of [624, 555, 23]-code), using the primitive BCH-code C(I) with length 624 = 54−1, defining interval I = {136,137,…,157}, and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(561, 624, F5, 19) (dual of [624, 563, 20]-code), using the primitive BCH-code C(I) with length 624 = 54−1, defining interval I = {138,139,…,156}, and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(51, 5, F5, 1) (dual of [5, 4, 2]-code), using
- Reed–Solomon code RS(4,5) [i]
- 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([136,156]), C2 = C([138,157]), C3 = C1 + C2 = C([138,156]), and C∩ = C1 ∩ C2 = C([136,157]) [i] based on
- discarding factors / shortening the dual code based on linear OA(570, 633, F5, 22) (dual of [633, 563, 23]-code), using
(48, 70, 34422)-Net in Base 5 — Upper bound on s
There is no (48, 70, 34423)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 8 470624 108451 375972 539198 054785 304718 311362 072373 > 570 [i]