Best Known (39, 39+38, s)-Nets in Base 32
(39, 39+38, 224)-Net over F32 — Constructive and digital
Digital (39, 77, 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, 49, 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
(39, 39+38, 288)-Net in Base 32 — Constructive
(39, 77, 288)-net in base 32, using
- 28 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
(39, 39+38, 677)-Net over F32 — Digital
Digital (39, 77, 677)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3277, 677, F32, 38) (dual of [677, 600, 39]-code), using
- discarding factors / shortening the dual code based on linear OA(3277, 1039, F32, 38) (dual of [1039, 962, 39]-code), using
- construction XX applied to C1 = C([1020,33]), C2 = C([3,34]), C3 = C1 + C2 = C([3,33]), and C∩ = C1 ∩ C2 = C([1020,34]) [i] based on
- linear OA(3270, 1023, F32, 37) (dual of [1023, 953, 38]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−3,−2,…,33}, and designed minimum distance d ≥ |I|+1 = 38 [i]
- linear OA(3263, 1023, F32, 32) (dual of [1023, 960, 33]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {3,4,…,34}, and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(3272, 1023, F32, 38) (dual of [1023, 951, 39]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−3,−2,…,34}, and designed minimum distance d ≥ |I|+1 = 39 [i]
- linear OA(3261, 1023, F32, 31) (dual of [1023, 962, 32]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {3,4,…,33}, and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(325, 14, F32, 5) (dual of [14, 9, 6]-code or 14-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
- linear OA(320, 2, F32, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(320, s, F32, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction XX applied to C1 = C([1020,33]), C2 = C([3,34]), C3 = C1 + C2 = C([3,33]), and C∩ = C1 ∩ C2 = C([1020,34]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3277, 1039, F32, 38) (dual of [1039, 962, 39]-code), using
(39, 39+38, 321854)-Net in Base 32 — Upper bound on s
There is no (39, 77, 321855)-net in base 32, because
- 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]