Best Known (221−31, 221, s)-Nets in Base 4
(221−31, 221, 17480)-Net over F4 — Constructive and digital
Digital (190, 221, 17480)-net over F4, using
- 41 times duplication [i] based on digital (189, 220, 17480)-net over F4, using
- net defined by OOA [i] based on linear OOA(4220, 17480, F4, 31, 31) (dual of [(17480, 31), 541660, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(4220, 262201, F4, 31) (dual of [262201, 261981, 32]-code), using
- 2 times code embedding in larger space [i] based on linear OA(4218, 262199, F4, 31) (dual of [262199, 261981, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(24) [i] based on
- linear OA(4208, 262144, F4, 31) (dual of [262144, 261936, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(4163, 262144, F4, 25) (dual of [262144, 261981, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(410, 55, F4, 5) (dual of [55, 45, 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(30) ⊂ Ce(24) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(4218, 262199, F4, 31) (dual of [262199, 261981, 32]-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(4220, 262201, F4, 31) (dual of [262201, 261981, 32]-code), using
- net defined by OOA [i] based on linear OOA(4220, 17480, F4, 31, 31) (dual of [(17480, 31), 541660, 32]-NRT-code), using
(221−31, 221, 143642)-Net over F4 — Digital
Digital (190, 221, 143642)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4221, 143642, F4, 31) (dual of [143642, 143421, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(4221, 262202, F4, 31) (dual of [262202, 261981, 32]-code), using
- 3 times code embedding in larger space [i] based on linear OA(4218, 262199, F4, 31) (dual of [262199, 261981, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(24) [i] based on
- linear OA(4208, 262144, F4, 31) (dual of [262144, 261936, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(4163, 262144, F4, 25) (dual of [262144, 261981, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(410, 55, F4, 5) (dual of [55, 45, 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(30) ⊂ Ce(24) [i] based on
- 3 times code embedding in larger space [i] based on linear OA(4218, 262199, F4, 31) (dual of [262199, 261981, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(4221, 262202, F4, 31) (dual of [262202, 261981, 32]-code), using
(221−31, 221, large)-Net in Base 4 — Upper bound on s
There is no (190, 221, large)-net in base 4, because
- 29 times m-reduction [i] would yield (190, 192, large)-net in base 4, but