Best Known (104−51, 104, s)-Nets in Base 32
(104−51, 104, 240)-Net over F32 — Constructive and digital
Digital (53, 104, 240)-net over F32, using
- t-expansion [i] based on digital (51, 104, 240)-net over F32, using
- 5 times m-reduction [i] based on digital (51, 109, 240)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (11, 40, 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, 69, 120)-net over F32, using
- net from sequence [i] based on digital (11, 119)-sequence over F32 (see above)
- digital (11, 40, 120)-net over F32, using
- (u, u+v)-construction [i] based on
- 5 times m-reduction [i] based on digital (51, 109, 240)-net over F32, using
(104−51, 104, 513)-Net in Base 32 — Constructive
(53, 104, 513)-net in base 32, using
- t-expansion [i] based on (46, 104, 513)-net in base 32, using
- 4 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
- 4 times m-reduction [i] based on (46, 108, 513)-net in base 32, using
(104−51, 104, 877)-Net over F32 — Digital
Digital (53, 104, 877)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(32104, 877, F32, 51) (dual of [877, 773, 52]-code), using
- discarding factors / shortening the dual code based on linear OA(32104, 1036, F32, 51) (dual of [1036, 932, 52]-code), using
- construction X applied to C([0,25]) ⊂ C([0,23]) [i] based on
- linear OA(32101, 1025, F32, 51) (dual of [1025, 924, 52]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025 | 324−1, defining interval I = [0,25], and minimum distance d ≥ |{−25,−24,…,25}|+1 = 52 (BCH-bound) [i]
- linear OA(3293, 1025, F32, 47) (dual of [1025, 932, 48]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025 | 324−1, defining interval I = [0,23], and minimum distance d ≥ |{−23,−22,…,23}|+1 = 48 (BCH-bound) [i]
- linear OA(323, 11, F32, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,32) or 11-cap in PG(2,32)), using
- discarding factors / shortening the dual code based on linear OA(323, 32, F32, 3) (dual of [32, 29, 4]-code or 32-arc in PG(2,32) or 32-cap in PG(2,32)), using
- Reed–Solomon code RS(29,32) [i]
- discarding factors / shortening the dual code based on linear OA(323, 32, F32, 3) (dual of [32, 29, 4]-code or 32-arc in PG(2,32) or 32-cap in PG(2,32)), using
- construction X applied to C([0,25]) ⊂ C([0,23]) [i] based on
- discarding factors / shortening the dual code based on linear OA(32104, 1036, F32, 51) (dual of [1036, 932, 52]-code), using
(104−51, 104, 521760)-Net in Base 32 — Upper bound on s
There is no (53, 104, 521761)-net in base 32, because
- 1 times m-reduction [i] would yield (53, 103, 521761)-net in base 32, but
- the generalized Rao bound for nets shows that 32m ≥ 107263 878989 062173 887871 446699 390946 167190 700729 352288 243161 013904 478462 415542 050981 433077 650278 041819 327710 730380 511308 272517 995323 177381 908242 145443 455392 > 32103 [i]