Best Known (111, 111+21, s)-Nets in Base 4
(111, 111+21, 6562)-Net over F4 — Constructive and digital
Digital (111, 132, 6562)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (1, 11, 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 (100, 121, 6553)-net over F4, using
- net defined by OOA [i] based on linear OOA(4121, 6553, F4, 21, 21) (dual of [(6553, 21), 137492, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(4121, 65531, F4, 21) (dual of [65531, 65410, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(4121, 65536, F4, 21) (dual of [65536, 65415, 22]-code), using
- an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- discarding factors / shortening the dual code based on linear OA(4121, 65536, F4, 21) (dual of [65536, 65415, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(4121, 65531, F4, 21) (dual of [65531, 65410, 22]-code), using
- net defined by OOA [i] based on linear OOA(4121, 6553, F4, 21, 21) (dual of [(6553, 21), 137492, 22]-NRT-code), using
- digital (1, 11, 9)-net over F4, using
(111, 111+21, 37408)-Net over F4 — Digital
Digital (111, 132, 37408)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4132, 37408, F4, 21) (dual of [37408, 37276, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(4132, 65579, F4, 21) (dual of [65579, 65447, 22]-code), using
- 1 times code embedding in larger space [i] based on linear OA(4131, 65578, F4, 21) (dual of [65578, 65447, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(14) [i] based on
- linear OA(4121, 65536, F4, 21) (dual of [65536, 65415, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(489, 65536, F4, 15) (dual of [65536, 65447, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(410, 42, F4, 5) (dual of [42, 32, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(410, 63, F4, 5) (dual of [63, 53, 6]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 6 [i]
- discarding factors / shortening the dual code based on linear OA(410, 63, F4, 5) (dual of [63, 53, 6]-code), using
- construction X applied to Ce(20) ⊂ Ce(14) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(4131, 65578, F4, 21) (dual of [65578, 65447, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(4132, 65579, F4, 21) (dual of [65579, 65447, 22]-code), using
(111, 111+21, large)-Net in Base 4 — Upper bound on s
There is no (111, 132, large)-net in base 4, because
- 19 times m-reduction [i] would yield (111, 113, large)-net in base 4, but