Best Known (78−39, 78, s)-Nets in Base 32
(78−39, 78, 224)-Net over F32 — Constructive and digital
Digital (39, 78, 224)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (9, 28, 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, 50, 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, 28, 104)-net over F32, using
(78−39, 78, 288)-Net in Base 32 — Constructive
(39, 78, 288)-net in base 32, using
- 27 times m-reduction [i] based on (39, 105, 288)-net in base 32, using
- base change [i] based on digital (9, 75, 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, 75, 288)-net over F128, using
(78−39, 78, 624)-Net over F32 — Digital
Digital (39, 78, 624)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3278, 624, F32, 39) (dual of [624, 546, 40]-code), using
- discarding factors / shortening the dual code based on linear OA(3278, 1030, F32, 39) (dual of [1030, 952, 40]-code), using
- construction X applied to C([0,19]) ⊂ C([0,18]) [i] based on
- 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(3273, 1025, F32, 37) (dual of [1025, 952, 38]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025 | 324−1, defining interval I = [0,18], and minimum distance d ≥ |{−18,−17,…,18}|+1 = 38 (BCH-bound) [i]
- linear OA(321, 5, F32, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(321, s, F32, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,19]) ⊂ C([0,18]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3278, 1030, F32, 39) (dual of [1030, 952, 40]-code), using
(78−39, 78, 321854)-Net in Base 32 — Upper bound on s
There is no (39, 78, 321855)-net in base 32, because
- 1 times m-reduction [i] would yield (39, 77, 321855)-net in base 32, but
- the generalized Rao bound for nets shows that 32m ≥ 78 807754 498573 041363 419663 588296 285729 607989 023595 328872 327182 880774 725102 795992 003783 921638 883966 512836 551890 863428 > 3277 [i]