Best Known (8, 18, s)-Nets in Base 16
(8, 18, 65)-Net over F16 — Constructive and digital
Digital (8, 18, 65)-net over F16, using
- t-expansion [i] based on digital (6, 18, 65)-net over F16, using
- net from sequence [i] based on digital (6, 64)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 6 and N(F) ≥ 65, using
- the Hermitian function field over F16 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 6 and N(F) ≥ 65, using
- net from sequence [i] based on digital (6, 64)-sequence over F16, using
(8, 18, 88)-Net over F16 — Digital
Digital (8, 18, 88)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1618, 88, F16, 10) (dual of [88, 70, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(1618, 89, F16, 10) (dual of [89, 71, 11]-code), using
- construction XX applied to C1 = C([4,12]), C2 = C([3,11]), C3 = C1 + C2 = C([4,11]), and C∩ = C1 ∩ C2 = C([3,12]) [i] based on
- linear OA(1616, 85, F16, 9) (dual of [85, 69, 10]-code), using the BCH-code C(I) with length 85 | 162−1, defining interval I = {4,5,…,12}, and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(1616, 85, F16, 9) (dual of [85, 69, 10]-code), using the BCH-code C(I) with length 85 | 162−1, defining interval I = {3,4,…,11}, and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(1618, 85, F16, 10) (dual of [85, 67, 11]-code), using the BCH-code C(I) with length 85 | 162−1, defining interval I = {3,4,…,12}, and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(1614, 85, F16, 8) (dual of [85, 71, 9]-code), using the BCH-code C(I) with length 85 | 162−1, defining interval I = {4,5,…,11}, and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(160, 2, F16, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(160, s, F16, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(160, 2, F16, 0) (dual of [2, 2, 1]-code) (see above)
- construction XX applied to C1 = C([4,12]), C2 = C([3,11]), C3 = C1 + C2 = C([4,11]), and C∩ = C1 ∩ C2 = C([3,12]) [i] based on
- discarding factors / shortening the dual code based on linear OA(1618, 89, F16, 10) (dual of [89, 71, 11]-code), using
(8, 18, 129)-Net in Base 16 — Constructive
(8, 18, 129)-net in base 16, using
- base change [i] based on (2, 12, 129)-net in base 64, using
- 2 times m-reduction [i] based on (2, 14, 129)-net in base 64, using
- base change [i] based on digital (0, 12, 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, 12, 129)-net over F128, using
- 2 times m-reduction [i] based on (2, 14, 129)-net in base 64, using
(8, 18, 3752)-Net in Base 16 — Upper bound on s
There is no (8, 18, 3753)-net in base 16, because
- the generalized Rao bound for nets shows that 16m ≥ 4726 256054 195595 219976 > 1618 [i]