Best Known (33, 33+31, s)-Nets in Base 32
(33, 33+31, 218)-Net over F32 — Constructive and digital
Digital (33, 64, 218)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (7, 22, 98)-net over F32, using
- net from sequence [i] based on digital (7, 97)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 7 and N(F) ≥ 98, using
- net from sequence [i] based on digital (7, 97)-sequence over F32, using
- digital (11, 42, 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 (7, 22, 98)-net over F32, using
(33, 33+31, 288)-Net in Base 32 — Constructive
(33, 64, 288)-net in base 32, using
- 20 times m-reduction [i] based on (33, 84, 288)-net in base 32, using
- base change [i] based on digital (9, 60, 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, 60, 288)-net over F128, using
(33, 33+31, 687)-Net over F32 — Digital
Digital (33, 64, 687)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3264, 687, F32, 31) (dual of [687, 623, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(3264, 1036, F32, 31) (dual of [1036, 972, 32]-code), using
- construction X applied to C([0,15]) ⊂ C([0,13]) [i] based on
- 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(3253, 1025, F32, 27) (dual of [1025, 972, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025 | 324−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (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([0,15]) ⊂ C([0,13]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3264, 1036, F32, 31) (dual of [1036, 972, 32]-code), using
(33, 33+31, 434537)-Net in Base 32 — Upper bound on s
There is no (33, 64, 434538)-net in base 32, because
- 1 times m-reduction [i] would yield (33, 63, 434538)-net in base 32, but
- the generalized Rao bound for nets shows that 32m ≥ 66750 482204 535904 858271 325648 182721 696314 471021 050431 511893 349549 768767 772533 298956 701848 797328 > 3263 [i]