Best Known (68−32, 68, s)-Nets in Base 32
(68−32, 68, 224)-Net over F32 — Constructive and digital
Digital (36, 68, 224)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (9, 25, 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, 43, 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, 25, 104)-net over F32, using
(68−32, 68, 288)-Net in Base 32 — Constructive
(36, 68, 288)-net in base 32, using
- t-expansion [i] based on (35, 68, 288)-net in base 32, using
- 23 times m-reduction [i] based on (35, 91, 288)-net in base 32, using
- base change [i] based on digital (9, 65, 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, 65, 288)-net over F128, using
- 23 times m-reduction [i] based on (35, 91, 288)-net in base 32, using
(68−32, 68, 879)-Net over F32 — Digital
Digital (36, 68, 879)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3268, 879, F32, 32) (dual of [879, 811, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(3268, 1042, F32, 32) (dual of [1042, 974, 33]-code), using
- construction XX applied to C1 = C([1017,24]), C2 = C([0,25]), C3 = C1 + C2 = C([0,24]), and C∩ = C1 ∩ C2 = C([1017,25]) [i] based on
- 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 = {−6,−5,…,24}, and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(3251, 1023, F32, 26) (dual of [1023, 972, 27]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,25], and designed minimum distance d ≥ |I|+1 = 27 [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 = {−6,−5,…,25}, and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(3249, 1023, F32, 25) (dual of [1023, 974, 26]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,24], and designed minimum distance d ≥ |I|+1 = 26 [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
- 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([1017,24]), C2 = C([0,25]), C3 = C1 + C2 = C([0,24]), and C∩ = C1 ∩ C2 = C([1017,25]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3268, 1042, F32, 32) (dual of [1042, 974, 33]-code), using
(68−32, 68, 547089)-Net in Base 32 — Upper bound on s
There is no (36, 68, 547090)-net in base 32, because
- the generalized Rao bound for nets shows that 32m ≥ 2 239802 610579 455632 957094 142230 567424 450619 863577 508129 846274 387297 570869 707697 065308 396598 308540 700353 > 3268 [i]