Best Known (127−31, 127, s)-Nets in Base 5
(127−31, 127, 408)-Net over F5 — Constructive and digital
Digital (96, 127, 408)-net over F5, using
- 5 times m-reduction [i] based on digital (96, 132, 408)-net over F5, using
- trace code for nets [i] based on digital (30, 66, 204)-net over F25, using
- net from sequence [i] based on digital (30, 203)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 30 and N(F) ≥ 204, using
- net from sequence [i] based on digital (30, 203)-sequence over F25, using
- trace code for nets [i] based on digital (30, 66, 204)-net over F25, using
(127−31, 127, 3156)-Net over F5 — Digital
Digital (96, 127, 3156)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5127, 3156, F5, 31) (dual of [3156, 3029, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(5127, 3164, F5, 31) (dual of [3164, 3037, 32]-code), using
- 32 step Varšamov–Edel lengthening with (ri) = (3, 0, 1, 0, 0, 0, 1, 7 times 0, 1, 17 times 0) [i] based on linear OA(5121, 3126, F5, 31) (dual of [3126, 3005, 32]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 3126 | 510−1, defining interval I = [0,15], and minimum distance d ≥ |{−15,−14,…,15}|+1 = 32 (BCH-bound) [i]
- 32 step Varšamov–Edel lengthening with (ri) = (3, 0, 1, 0, 0, 0, 1, 7 times 0, 1, 17 times 0) [i] based on linear OA(5121, 3126, F5, 31) (dual of [3126, 3005, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(5127, 3164, F5, 31) (dual of [3164, 3037, 32]-code), using
(127−31, 127, 1194127)-Net in Base 5 — Upper bound on s
There is no (96, 127, 1194128)-net in base 5, because
- 1 times m-reduction [i] would yield (96, 126, 1194128)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 11754 986402 313877 917490 929290 577619 464679 575564 327420 328151 171451 582423 729846 693903 772225 > 5126 [i]