Best Known (33−5, 33, s)-Nets in Base 4
(33−5, 33, 524294)-Net over F4 — Constructive and digital
Digital (28, 33, 524294)-net over F4, using
- net defined by OOA [i] based on linear OOA(433, 524294, F4, 5, 5) (dual of [(524294, 5), 2621437, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(433, 1048589, F4, 5) (dual of [1048589, 1048556, 6]-code), using
- 1 times code embedding in larger space [i] based on linear OA(432, 1048588, F4, 5) (dual of [1048588, 1048556, 6]-code), using
- construction X4 applied to Ce(4) ⊂ Ce(2) [i] based on
- linear OA(431, 1048576, F4, 5) (dual of [1048576, 1048545, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(421, 1048576, F4, 3) (dual of [1048576, 1048555, 4]-code or 1048576-cap in PG(20,4)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(411, 12, F4, 11) (dual of [12, 1, 12]-code or 12-arc in PG(10,4)), using
- dual of repetition code with length 12 [i]
- linear OA(41, 12, F4, 1) (dual of [12, 11, 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
- construction X4 applied to Ce(4) ⊂ Ce(2) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(432, 1048588, F4, 5) (dual of [1048588, 1048556, 6]-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(433, 1048589, F4, 5) (dual of [1048589, 1048556, 6]-code), using
(33−5, 33, 1048590)-Net over F4 — Digital
Digital (28, 33, 1048590)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(433, 1048590, F4, 5) (dual of [1048590, 1048557, 6]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(432, 1048588, F4, 5) (dual of [1048588, 1048556, 6]-code), using
- construction X4 applied to Ce(4) ⊂ Ce(2) [i] based on
- linear OA(431, 1048576, F4, 5) (dual of [1048576, 1048545, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(421, 1048576, F4, 3) (dual of [1048576, 1048555, 4]-code or 1048576-cap in PG(20,4)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(411, 12, F4, 11) (dual of [12, 1, 12]-code or 12-arc in PG(10,4)), using
- dual of repetition code with length 12 [i]
- linear OA(41, 12, F4, 1) (dual of [12, 11, 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
- construction X4 applied to Ce(4) ⊂ Ce(2) [i] based on
- linear OA(432, 1048589, F4, 4) (dual of [1048589, 1048557, 5]-code), using Gilbert–Varšamov bound and bm = 432 > Vbs−1(k−1) = 5 188314 997829 671339 [i]
- linear OA(40, 1, F4, 0) (dual of [1, 1, 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
- linear OA(432, 1048588, F4, 5) (dual of [1048588, 1048556, 6]-code), using
- construction X with Varšamov bound [i] based on
(33−5, 33, large)-Net in Base 4 — Upper bound on s
There is no (28, 33, large)-net in base 4, because
- 3 times m-reduction [i] would yield (28, 30, large)-net in base 4, but