Best Known (53, 53+49, s)-Nets in Base 32
(53, 53+49, 240)-Net over F32 — Constructive and digital
Digital (53, 102, 240)-net over F32, using
- t-expansion [i] based on digital (51, 102, 240)-net over F32, using
- 7 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
- 7 times m-reduction [i] based on digital (51, 109, 240)-net over F32, using
(53, 53+49, 513)-Net in Base 32 — Constructive
(53, 102, 513)-net in base 32, using
- t-expansion [i] based on (46, 102, 513)-net in base 32, using
- 6 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
- 6 times m-reduction [i] based on (46, 108, 513)-net in base 32, using
(53, 53+49, 995)-Net over F32 — Digital
Digital (53, 102, 995)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(32102, 995, F32, 49) (dual of [995, 893, 50]-code), using
- discarding factors / shortening the dual code based on linear OA(32102, 1042, F32, 49) (dual of [1042, 940, 50]-code), using
- construction X applied to C([0,24]) ⊂ C([0,21]) [i] based on
- linear OA(3297, 1025, F32, 49) (dual of [1025, 928, 50]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025 | 324−1, defining interval I = [0,24], and minimum distance d ≥ |{−24,−23,…,24}|+1 = 50 (BCH-bound) [i]
- linear OA(3285, 1025, F32, 43) (dual of [1025, 940, 44]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025 | 324−1, defining interval I = [0,21], and minimum distance d ≥ |{−21,−20,…,21}|+1 = 44 (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([0,24]) ⊂ C([0,21]) [i] based on
- discarding factors / shortening the dual code based on linear OA(32102, 1042, F32, 49) (dual of [1042, 940, 50]-code), using
(53, 53+49, 682599)-Net in Base 32 — Upper bound on s
There is no (53, 102, 682600)-net in base 32, because
- 1 times m-reduction [i] would yield (53, 101, 682600)-net in base 32, but
- the generalized Rao bound for nets shows that 32m ≥ 104 749803 458779 252875 068147 818790 977305 799710 633832 974616 496312 418433 758358 975793 187588 738809 795835 617674 383224 727408 531789 294169 838477 917917 947039 935756 > 32101 [i]