Best Known (9, 9+11, s)-Nets in Base 16
(9, 9+11, 66)-Net over F16 — Constructive and digital
Digital (9, 20, 66)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (2, 7, 120)-net over F16, using
- net defined by OOA [i] based on linear OOA(167, 120, F16, 5, 5) (dual of [(120, 5), 593, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(167, 241, F16, 5) (dual of [241, 234, 6]-code), using
- net defined by OOA [i] based on linear OOA(167, 120, F16, 5, 5) (dual of [(120, 5), 593, 6]-NRT-code), using
- digital (2, 13, 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 (2, 7, 120)-net over F16, using
(9, 9+11, 89)-Net over F16 — Digital
Digital (9, 20, 89)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1620, 89, F16, 11) (dual of [89, 69, 12]-code), using
- construction XX applied to C1 = C([3,12]), C2 = C([2,11]), C3 = C1 + C2 = C([3,11]), and C∩ = C1 ∩ C2 = C([2,12]) [i] based on
- 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(1618, 85, F16, 10) (dual of [85, 67, 11]-code), using the BCH-code C(I) with length 85 | 162−1, defining interval I = {2,3,…,11}, and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(1620, 85, F16, 11) (dual of [85, 65, 12]-code), using the BCH-code C(I) with length 85 | 162−1, defining interval I = {2,3,…,12}, and designed minimum distance d ≥ |I|+1 = 12 [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(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([3,12]), C2 = C([2,11]), C3 = C1 + C2 = C([3,11]), and C∩ = C1 ∩ C2 = C([2,12]) [i] based on
(9, 9+11, 129)-Net in Base 16 — Constructive
(9, 20, 129)-net in base 16, using
- 1 times m-reduction [i] based on (9, 21, 129)-net in base 16, 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
(9, 9+11, 6535)-Net in Base 16 — Upper bound on s
There is no (9, 20, 6536)-net in base 16, because
- 1 times m-reduction [i] would yield (9, 19, 6536)-net in base 16, but
- the generalized Rao bound for nets shows that 16m ≥ 75615 399506 297683 797451 > 1619 [i]