Best Known (109, 109+23, s)-Nets in Base 4
(109, 109+23, 1499)-Net over F4 — Constructive and digital
Digital (109, 132, 1499)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (1, 12, 9)-net over F4, using
- net from sequence [i] based on digital (1, 8)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 1 and N(F) ≥ 9, using
- the Hermitian function field over F4 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 1 and N(F) ≥ 9, using
- net from sequence [i] based on digital (1, 8)-sequence over F4, using
- digital (97, 120, 1490)-net over F4, using
- net defined by OOA [i] based on linear OOA(4120, 1490, F4, 23, 23) (dual of [(1490, 23), 34150, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(4120, 16391, F4, 23) (dual of [16391, 16271, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- linear OA(4120, 16384, F4, 23) (dual of [16384, 16264, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(4113, 16384, F4, 22) (dual of [16384, 16271, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(40, 7, F4, 0) (dual of [7, 7, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(40, s, F4, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- OOA 11-folding and stacking with additional row [i] based on linear OA(4120, 16391, F4, 23) (dual of [16391, 16271, 24]-code), using
- net defined by OOA [i] based on linear OOA(4120, 1490, F4, 23, 23) (dual of [(1490, 23), 34150, 24]-NRT-code), using
- digital (1, 12, 9)-net over F4, using
(109, 109+23, 16433)-Net over F4 — Digital
Digital (109, 132, 16433)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4132, 16433, F4, 23) (dual of [16433, 16301, 24]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(4129, 16428, F4, 23) (dual of [16428, 16299, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(16) [i] based on
- linear OA(4120, 16384, F4, 23) (dual of [16384, 16264, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(485, 16384, F4, 17) (dual of [16384, 16299, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(49, 44, F4, 5) (dual of [44, 35, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(49, 51, F4, 5) (dual of [51, 42, 6]-code), using
- a “DaH†code from Brouwer’s database [i]
- discarding factors / shortening the dual code based on linear OA(49, 51, F4, 5) (dual of [51, 42, 6]-code), using
- construction X applied to Ce(22) ⊂ Ce(16) [i] based on
- linear OA(4129, 16430, F4, 21) (dual of [16430, 16301, 22]-code), using Gilbert–Varšamov bound and bm = 4129 > Vbs−1(k−1) = 2908 539522 302446 317784 341388 406326 916759 679207 412129 264808 379326 794328 877166 [i]
- linear OA(41, 3, F4, 1) (dual of [3, 2, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- linear OA(4129, 16428, F4, 23) (dual of [16428, 16299, 24]-code), using
- construction X with Varšamov bound [i] based on
(109, 109+23, large)-Net in Base 4 — Upper bound on s
There is no (109, 132, large)-net in base 4, because
- 21 times m-reduction [i] would yield (109, 111, large)-net in base 4, but