Best Known (69−20, 69, s)-Nets in Base 5
(69−20, 69, 252)-Net over F5 — Constructive and digital
Digital (49, 69, 252)-net over F5, using
- 9 times m-reduction [i] based on digital (49, 78, 252)-net over F5, using
- trace code for nets [i] based on digital (10, 39, 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, 39, 126)-net over F25, using
(69−20, 69, 706)-Net over F5 — Digital
Digital (49, 69, 706)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(569, 706, F5, 20) (dual of [706, 637, 21]-code), using
- 76 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 10 times 0, 1, 22 times 0, 1, 38 times 0) [i] based on linear OA(564, 625, F5, 20) (dual of [625, 561, 21]-code), using
- 1 times truncation [i] based on linear OA(565, 626, F5, 21) (dual of [626, 561, 22]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 626 | 58−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- 1 times truncation [i] based on linear OA(565, 626, F5, 21) (dual of [626, 561, 22]-code), using
- 76 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 10 times 0, 1, 22 times 0, 1, 38 times 0) [i] based on linear OA(564, 625, F5, 20) (dual of [625, 561, 21]-code), using
(69−20, 69, 75295)-Net in Base 5 — Upper bound on s
There is no (49, 69, 75296)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 1 694180 390832 307777 997412 930322 258988 437759 132161 > 569 [i]