Best Known (125−31, 125, s)-Nets in Base 5
(125−31, 125, 408)-Net over F5 — Constructive and digital
Digital (94, 125, 408)-net over F5, using
- 3 times m-reduction [i] based on digital (94, 128, 408)-net over F5, using
- trace code for nets [i] based on digital (30, 64, 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, 64, 204)-net over F25, using
(125−31, 125, 2823)-Net over F5 — Digital
Digital (94, 125, 2823)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5125, 2823, F5, 31) (dual of [2823, 2698, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(5125, 3140, F5, 31) (dual of [3140, 3015, 32]-code), using
- construction X applied to C([0,15]) ⊂ C([0,13]) [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]
- linear OA(5111, 3126, F5, 27) (dual of [3126, 3015, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 3126 | 510−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(54, 14, F5, 3) (dual of [14, 10, 4]-code or 14-cap in PG(3,5)), using
- construction X applied to C([0,15]) ⊂ C([0,13]) [i] based on
- discarding factors / shortening the dual code based on linear OA(5125, 3140, F5, 31) (dual of [3140, 3015, 32]-code), using
(125−31, 125, 963504)-Net in Base 5 — Upper bound on s
There is no (94, 125, 963505)-net in base 5, because
- 1 times m-reduction [i] would yield (94, 124, 963505)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 470 199898 910496 742827 281131 797901 858980 476219 455567 114699 947543 815092 328242 557332 539725 > 5124 [i]