Best Known (11, 11+13, s)-Nets in Base 16
(11, 11+13, 71)-Net over F16 — Constructive and digital
Digital (11, 24, 71)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (2, 8, 33)-net over F16, using
- net from sequence [i] based on digital (2, 32)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 2 and N(F) ≥ 33, using
- net from sequence [i] based on digital (2, 32)-sequence over F16, using
- digital (3, 16, 38)-net over F16, using
- net from sequence [i] based on digital (3, 37)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 3 and N(F) ≥ 38, using
- net from sequence [i] based on digital (3, 37)-sequence over F16, using
- digital (2, 8, 33)-net over F16, using
(11, 11+13, 98)-Net over F16 — Digital
Digital (11, 24, 98)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1624, 98, F16, 13) (dual of [98, 74, 14]-code), using
- 8 step Varšamov–Edel lengthening with (ri) = (1, 7 times 0) [i] based on linear OA(1623, 89, F16, 13) (dual of [89, 66, 14]-code), using
- construction XX applied to C1 = C([12,23]), C2 = C([11,22]), C3 = C1 + C2 = C([12,22]), and C∩ = C1 ∩ C2 = C([11,23]) [i] based on
- linear OA(1621, 85, F16, 12) (dual of [85, 64, 13]-code), using the BCH-code C(I) with length 85 | 162−1, defining interval I = {12,13,…,23}, and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(1621, 85, F16, 12) (dual of [85, 64, 13]-code), using the BCH-code C(I) with length 85 | 162−1, defining interval I = {11,12,…,22}, and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(1623, 85, F16, 13) (dual of [85, 62, 14]-code), using the BCH-code C(I) with length 85 | 162−1, defining interval I = {11,12,…,23}, and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(1619, 85, F16, 11) (dual of [85, 66, 12]-code), using the BCH-code C(I) with length 85 | 162−1, defining interval I = {12,13,…,22}, and designed minimum distance d ≥ |I|+1 = 12 [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([12,23]), C2 = C([11,22]), C3 = C1 + C2 = C([12,22]), and C∩ = C1 ∩ C2 = C([11,23]) [i] based on
- 8 step Varšamov–Edel lengthening with (ri) = (1, 7 times 0) [i] based on linear OA(1623, 89, F16, 13) (dual of [89, 66, 14]-code), using
(11, 11+13, 129)-Net in Base 16 — Constructive
(11, 24, 129)-net in base 16, using
- base change [i] based on (3, 16, 129)-net in base 64, using
- 5 times m-reduction [i] based on (3, 21, 129)-net in base 64, using
- base change [i] based on digital (0, 18, 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, 18, 129)-net over F128, using
- 5 times m-reduction [i] based on (3, 21, 129)-net in base 64, using
(11, 11+13, 8237)-Net in Base 16 — Upper bound on s
There is no (11, 24, 8238)-net in base 16, because
- 1 times m-reduction [i] would yield (11, 23, 8238)-net in base 16, but
- the generalized Rao bound for nets shows that 16m ≥ 4955 218690 570698 916657 329796 > 1623 [i]