Best Known (37, 72, s)-Nets in Base 32
(37, 72, 224)-Net over F32 — Constructive and digital
Digital (37, 72, 224)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (9, 26, 104)-net over F32, using
- net from sequence [i] based on digital (9, 103)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 9 and N(F) ≥ 104, using
- net from sequence [i] based on digital (9, 103)-sequence over F32, using
- digital (11, 46, 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 (9, 26, 104)-net over F32, using
(37, 72, 288)-Net in Base 32 — Constructive
(37, 72, 288)-net in base 32, using
- 26 times m-reduction [i] based on (37, 98, 288)-net in base 32, using
- base change [i] based on digital (9, 70, 288)-net over F128, using
- net from sequence [i] based on digital (9, 287)-sequence over F128, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 9 and N(F) ≥ 288, using
- net from sequence [i] based on digital (9, 287)-sequence over F128, using
- base change [i] based on digital (9, 70, 288)-net over F128, using
(37, 72, 720)-Net over F32 — Digital
Digital (37, 72, 720)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3272, 720, F32, 35) (dual of [720, 648, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(3272, 1036, F32, 35) (dual of [1036, 964, 36]-code), using
- construction X applied to C([0,17]) ⊂ C([0,15]) [i] based on
- linear OA(3269, 1025, F32, 35) (dual of [1025, 956, 36]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025 | 324−1, defining interval I = [0,17], and minimum distance d ≥ |{−17,−16,…,17}|+1 = 36 (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(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([0,17]) ⊂ C([0,15]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3272, 1036, F32, 35) (dual of [1036, 964, 36]-code), using
(37, 72, 447496)-Net in Base 32 — Upper bound on s
There is no (37, 72, 447497)-net in base 32, because
- 1 times m-reduction [i] would yield (37, 71, 447497)-net in base 32, but
- the generalized Rao bound for nets shows that 32m ≥ 73392 805976 108137 059189 476956 666379 973998 338830 429957 588782 799365 282257 559632 966558 382380 002007 948414 704128 > 3271 [i]