Best Known (4, 4+6, s)-Nets in Base 32
(4, 4+6, 77)-Net over F32 — Constructive and digital
Digital (4, 10, 77)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (0, 3, 33)-net over F32, using
- net from sequence [i] based on digital (0, 32)-sequence over F32, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 0 and N(F) ≥ 33, using
- the rational function field F32(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 32)-sequence over F32, using
- digital (1, 7, 44)-net over F32, using
- net from sequence [i] based on digital (1, 43)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 1 and N(F) ≥ 44, using
- net from sequence [i] based on digital (1, 43)-sequence over F32, using
- digital (0, 3, 33)-net over F32, using
(4, 4+6, 97)-Net over F32 — Digital
Digital (4, 10, 97)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3210, 97, F32, 6) (dual of [97, 87, 7]-code), using
- construction XX applied to C1 = C([92,3]), C2 = C([0,4]), C3 = C1 + C2 = C([0,3]), and C∩ = C1 ∩ C2 = C([92,4]) [i] based on
- linear OA(328, 93, F32, 5) (dual of [93, 85, 6]-code), using the BCH-code C(I) with length 93 | 322−1, defining interval I = {−1,0,1,2,3}, and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(328, 93, F32, 5) (dual of [93, 85, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 93 | 322−1, defining interval I = [0,4], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(3210, 93, F32, 6) (dual of [93, 83, 7]-code), using the BCH-code C(I) with length 93 | 322−1, defining interval I = {−1,0,…,4}, and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(326, 93, F32, 4) (dual of [93, 87, 5]-code), using the expurgated narrow-sense BCH-code C(I) with length 93 | 322−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 5 [i]
- 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
- linear OA(320, 2, F32, 0) (dual of [2, 2, 1]-code) (see above)
- construction XX applied to C1 = C([92,3]), C2 = C([0,4]), C3 = C1 + C2 = C([0,3]), and C∩ = C1 ∩ C2 = C([92,4]) [i] based on
(4, 4+6, 129)-Net in Base 32 — Constructive
(4, 10, 129)-net in base 32, using
- 4 times m-reduction [i] based on (4, 14, 129)-net in base 32, using
- base change [i] based on digital (0, 10, 129)-net over F128, using
- net from sequence [i] based on digital (0, 128)-sequence over F128, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 0 and N(F) ≥ 129, using
- the rational function field F128(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 128)-sequence over F128, using
- base change [i] based on digital (0, 10, 129)-net over F128, using
(4, 4+6, 6096)-Net in Base 32 — Upper bound on s
There is no (4, 10, 6097)-net in base 32, because
- the generalized Rao bound for nets shows that 32m ≥ 1125 943900 014480 > 3210 [i]