Best Known (221−28, 221, s)-Nets in Base 4
(221−28, 221, 74902)-Net over F4 — Constructive and digital
Digital (193, 221, 74902)-net over F4, using
- net defined by OOA [i] based on linear OOA(4221, 74902, F4, 28, 28) (dual of [(74902, 28), 2097035, 29]-NRT-code), using
- OA 14-folding and stacking [i] based on linear OA(4221, 1048628, F4, 28) (dual of [1048628, 1048407, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(4221, 1048636, F4, 28) (dual of [1048636, 1048415, 29]-code), using
- construction X applied to Ce(28) ⊂ Ce(21) [i] based on
- linear OA(4211, 1048576, F4, 29) (dual of [1048576, 1048365, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(4161, 1048576, F4, 22) (dual of [1048576, 1048415, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(410, 60, F4, 5) (dual of [60, 50, 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(28) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(4221, 1048636, F4, 28) (dual of [1048636, 1048415, 29]-code), using
- OA 14-folding and stacking [i] based on linear OA(4221, 1048628, F4, 28) (dual of [1048628, 1048407, 29]-code), using
(221−28, 221, 524318)-Net over F4 — Digital
Digital (193, 221, 524318)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4221, 524318, F4, 2, 28) (dual of [(524318, 2), 1048415, 29]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4221, 1048636, F4, 28) (dual of [1048636, 1048415, 29]-code), using
- construction X applied to Ce(28) ⊂ Ce(21) [i] based on
- linear OA(4211, 1048576, F4, 29) (dual of [1048576, 1048365, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(4161, 1048576, F4, 22) (dual of [1048576, 1048415, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(410, 60, F4, 5) (dual of [60, 50, 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(28) ⊂ Ce(21) [i] based on
- OOA 2-folding [i] based on linear OA(4221, 1048636, F4, 28) (dual of [1048636, 1048415, 29]-code), using
(221−28, 221, large)-Net in Base 4 — Upper bound on s
There is no (193, 221, large)-net in base 4, because
- 26 times m-reduction [i] would yield (193, 195, large)-net in base 4, but