Best Known (94−45, 94, s)-Nets in Base 32
(94−45, 94, 240)-Net over F32 — Constructive and digital
Digital (49, 94, 240)-net over F32, using
- 9 times m-reduction [i] based on digital (49, 103, 240)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (11, 38, 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, 65, 120)-net over F32, using
- net from sequence [i] based on digital (11, 119)-sequence over F32 (see above)
- digital (11, 38, 120)-net over F32, using
- (u, u+v)-construction [i] based on
(94−45, 94, 513)-Net in Base 32 — Constructive
(49, 94, 513)-net in base 32, using
- t-expansion [i] based on (46, 94, 513)-net in base 32, using
- 14 times m-reduction [i] based on (46, 108, 513)-net in base 32, using
- base change [i] based on digital (28, 90, 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
- base change [i] based on digital (28, 90, 513)-net over F64, using
- 14 times m-reduction [i] based on (46, 108, 513)-net in base 32, using
(94−45, 94, 960)-Net over F32 — Digital
Digital (49, 94, 960)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3294, 960, F32, 45) (dual of [960, 866, 46]-code), using
- discarding factors / shortening the dual code based on linear OA(3294, 1042, F32, 45) (dual of [1042, 948, 46]-code), using
- construction X applied to C([0,22]) ⊂ C([0,19]) [i] based on
- linear OA(3289, 1025, F32, 45) (dual of [1025, 936, 46]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025 | 324−1, defining interval I = [0,22], and minimum distance d ≥ |{−22,−21,…,22}|+1 = 46 (BCH-bound) [i]
- linear OA(3277, 1025, F32, 39) (dual of [1025, 948, 40]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025 | 324−1, defining interval I = [0,19], and minimum distance d ≥ |{−19,−18,…,19}|+1 = 40 (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([0,22]) ⊂ C([0,19]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3294, 1042, F32, 45) (dual of [1042, 948, 46]-code), using
(94−45, 94, 673230)-Net in Base 32 — Upper bound on s
There is no (49, 94, 673231)-net in base 32, because
- 1 times m-reduction [i] would yield (49, 93, 673231)-net in base 32, but
- the generalized Rao bound for nets shows that 32m ≥ 95 270858 686152 089132 743745 698342 184061 654154 300221 452953 319298 589914 713817 366608 693101 278318 759820 600074 622310 484628 881796 894519 923978 189012 > 3293 [i]