Best Known (137−24, 137, s)-Nets in Base 4
(137−24, 137, 1369)-Net over F4 — Constructive and digital
Digital (113, 137, 1369)-net over F4, using
- net defined by OOA [i] based on linear OOA(4137, 1369, F4, 24, 24) (dual of [(1369, 24), 32719, 25]-NRT-code), using
- OA 12-folding and stacking [i] based on linear OA(4137, 16428, F4, 24) (dual of [16428, 16291, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(4137, 16429, F4, 24) (dual of [16429, 16292, 25]-code), using
- construction X applied to Ce(24) ⊂ Ce(17) [i] based on
- linear OA(4127, 16384, F4, 25) (dual of [16384, 16257, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(492, 16384, F4, 18) (dual of [16384, 16292, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(410, 45, F4, 5) (dual of [45, 35, 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(24) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(4137, 16429, F4, 24) (dual of [16429, 16292, 25]-code), using
- OA 12-folding and stacking [i] based on linear OA(4137, 16428, F4, 24) (dual of [16428, 16291, 25]-code), using
(137−24, 137, 15889)-Net over F4 — Digital
Digital (113, 137, 15889)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4137, 15889, F4, 24) (dual of [15889, 15752, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(4137, 16429, F4, 24) (dual of [16429, 16292, 25]-code), using
- construction X applied to Ce(24) ⊂ Ce(17) [i] based on
- linear OA(4127, 16384, F4, 25) (dual of [16384, 16257, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(492, 16384, F4, 18) (dual of [16384, 16292, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(410, 45, F4, 5) (dual of [45, 35, 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(24) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(4137, 16429, F4, 24) (dual of [16429, 16292, 25]-code), using
(137−24, 137, large)-Net in Base 4 — Upper bound on s
There is no (113, 137, large)-net in base 4, because
- 22 times m-reduction [i] would yield (113, 115, large)-net in base 4, but