Best Known (41, 41+37, s)-Nets in Base 32
(41, 41+37, 240)-Net over F32 — Constructive and digital
Digital (41, 78, 240)-net over F32, using
- 1 times m-reduction [i] based on digital (41, 79, 240)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (11, 30, 120)-net over F32, using
- net from sequence [i] based on digital (11, 119)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 11 and N(F) ≥ 120, using
- net from sequence [i] based on digital (11, 119)-sequence over F32, using
- digital (11, 49, 120)-net over F32, using
- net from sequence [i] based on digital (11, 119)-sequence over F32 (see above)
- digital (11, 30, 120)-net over F32, using
- (u, u+v)-construction [i] based on
(41, 41+37, 513)-Net in Base 32 — Constructive
(41, 78, 513)-net in base 32, using
- base change [i] based on digital (28, 65, 513)-net over F64, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- the Hermitian function field over F64 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
(41, 41+37, 902)-Net over F32 — Digital
Digital (41, 78, 902)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3278, 902, F32, 37) (dual of [902, 824, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(3278, 1042, F32, 37) (dual of [1042, 964, 38]-code), using
- construction X applied to C([0,18]) ⊂ C([0,15]) [i] based on
- linear OA(3273, 1025, F32, 37) (dual of [1025, 952, 38]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025 | 324−1, defining interval I = [0,18], and minimum distance d ≥ |{−18,−17,…,18}|+1 = 38 (BCH-bound) [i]
- linear OA(3261, 1025, F32, 31) (dual of [1025, 964, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025 | 324−1, defining interval I = [0,15], and minimum distance d ≥ |{−15,−14,…,15}|+1 = 32 (BCH-bound) [i]
- linear OA(325, 17, F32, 5) (dual of [17, 12, 6]-code or 17-arc in PG(4,32)), using
- discarding factors / shortening the dual code based on linear OA(325, 32, F32, 5) (dual of [32, 27, 6]-code or 32-arc in PG(4,32)), using
- Reed–Solomon code RS(27,32) [i]
- discarding factors / shortening the dual code based on linear OA(325, 32, F32, 5) (dual of [32, 27, 6]-code or 32-arc in PG(4,32)), using
- construction X applied to C([0,18]) ⊂ C([0,15]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3278, 1042, F32, 37) (dual of [1042, 964, 38]-code), using
(41, 41+37, 669051)-Net in Base 32 — Upper bound on s
There is no (41, 78, 669052)-net in base 32, because
- 1 times m-reduction [i] would yield (41, 77, 669052)-net in base 32, but
- the generalized Rao bound for nets shows that 32m ≥ 78 804911 612582 320262 900214 229139 932146 105118 667950 914014 112979 880519 746035 294196 563533 580976 100864 449244 286496 877755 > 3277 [i]