Best Known (240−34, 240, s)-Nets in Base 4
(240−34, 240, 15423)-Net over F4 — Constructive and digital
Digital (206, 240, 15423)-net over F4, using
- 43 times duplication [i] based on digital (203, 237, 15423)-net over F4, using
- net defined by OOA [i] based on linear OOA(4237, 15423, F4, 34, 34) (dual of [(15423, 34), 524145, 35]-NRT-code), using
- OA 17-folding and stacking [i] based on linear OA(4237, 262191, F4, 34) (dual of [262191, 261954, 35]-code), using
- 4 times code embedding in larger space [i] based on linear OA(4233, 262187, F4, 34) (dual of [262187, 261954, 35]-code), using
- construction X applied to Ce(33) ⊂ Ce(28) [i] based on
- linear OA(4226, 262144, F4, 34) (dual of [262144, 261918, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(4190, 262144, F4, 29) (dual of [262144, 261954, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(47, 43, F4, 4) (dual of [43, 36, 5]-code), using
- construction X applied to Ce(33) ⊂ Ce(28) [i] based on
- 4 times code embedding in larger space [i] based on linear OA(4233, 262187, F4, 34) (dual of [262187, 261954, 35]-code), using
- OA 17-folding and stacking [i] based on linear OA(4237, 262191, F4, 34) (dual of [262191, 261954, 35]-code), using
- net defined by OOA [i] based on linear OOA(4237, 15423, F4, 34, 34) (dual of [(15423, 34), 524145, 35]-NRT-code), using
(240−34, 240, 133755)-Net over F4 — Digital
Digital (206, 240, 133755)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4240, 133755, F4, 34) (dual of [133755, 133515, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(4240, 262203, F4, 34) (dual of [262203, 261963, 35]-code), using
- 1 times code embedding in larger space [i] based on linear OA(4239, 262202, F4, 34) (dual of [262202, 261963, 35]-code), using
- construction X applied to Ce(33) ⊂ Ce(26) [i] based on
- linear OA(4226, 262144, F4, 34) (dual of [262144, 261918, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(4181, 262144, F4, 27) (dual of [262144, 261963, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(413, 58, F4, 6) (dual of [58, 45, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(413, 63, F4, 6) (dual of [63, 50, 7]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- discarding factors / shortening the dual code based on linear OA(413, 63, F4, 6) (dual of [63, 50, 7]-code), using
- construction X applied to Ce(33) ⊂ Ce(26) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(4239, 262202, F4, 34) (dual of [262202, 261963, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(4240, 262203, F4, 34) (dual of [262203, 261963, 35]-code), using
(240−34, 240, large)-Net in Base 4 — Upper bound on s
There is no (206, 240, large)-net in base 4, because
- 32 times m-reduction [i] would yield (206, 208, large)-net in base 4, but