Best Known (255−38, 255, s)-Nets in Base 4
(255−38, 255, 13797)-Net over F4 — Constructive and digital
Digital (217, 255, 13797)-net over F4, using
- 42 times duplication [i] based on digital (215, 253, 13797)-net over F4, using
- net defined by OOA [i] based on linear OOA(4253, 13797, F4, 38, 38) (dual of [(13797, 38), 524033, 39]-NRT-code), using
- OA 19-folding and stacking [i] based on linear OA(4253, 262143, F4, 38) (dual of [262143, 261890, 39]-code), using
- discarding factors / shortening the dual code based on linear OA(4253, 262144, F4, 38) (dual of [262144, 261891, 39]-code), using
- an extension Ce(37) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,37], and designed minimum distance d ≥ |I|+1 = 38 [i]
- discarding factors / shortening the dual code based on linear OA(4253, 262144, F4, 38) (dual of [262144, 261891, 39]-code), using
- OA 19-folding and stacking [i] based on linear OA(4253, 262143, F4, 38) (dual of [262143, 261890, 39]-code), using
- net defined by OOA [i] based on linear OOA(4253, 13797, F4, 38, 38) (dual of [(13797, 38), 524033, 39]-NRT-code), using
(255−38, 255, 104246)-Net over F4 — Digital
Digital (217, 255, 104246)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4255, 104246, F4, 2, 38) (dual of [(104246, 2), 208237, 39]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4255, 131077, F4, 2, 38) (dual of [(131077, 2), 261899, 39]-NRT-code), using
- 41 times duplication [i] based on linear OOA(4254, 131077, F4, 2, 38) (dual of [(131077, 2), 261900, 39]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4254, 262154, F4, 38) (dual of [262154, 261900, 39]-code), using
- 1 times code embedding in larger space [i] based on linear OA(4253, 262153, F4, 38) (dual of [262153, 261900, 39]-code), using
- construction X applied to Ce(37) ⊂ Ce(36) [i] based on
- linear OA(4253, 262144, F4, 38) (dual of [262144, 261891, 39]-code), using an extension Ce(37) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,37], and designed minimum distance d ≥ |I|+1 = 38 [i]
- linear OA(4244, 262144, F4, 37) (dual of [262144, 261900, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(40, 9, F4, 0) (dual of [9, 9, 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
- construction X applied to Ce(37) ⊂ Ce(36) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(4253, 262153, F4, 38) (dual of [262153, 261900, 39]-code), using
- OOA 2-folding [i] based on linear OA(4254, 262154, F4, 38) (dual of [262154, 261900, 39]-code), using
- 41 times duplication [i] based on linear OOA(4254, 131077, F4, 2, 38) (dual of [(131077, 2), 261900, 39]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4255, 131077, F4, 2, 38) (dual of [(131077, 2), 261899, 39]-NRT-code), using
(255−38, 255, large)-Net in Base 4 — Upper bound on s
There is no (217, 255, large)-net in base 4, because
- 36 times m-reduction [i] would yield (217, 219, large)-net in base 4, but