Best Known (67−12, 67, s)-Nets in Base 4
(67−12, 67, 2734)-Net over F4 — Constructive and digital
Digital (55, 67, 2734)-net over F4, using
- net defined by OOA [i] based on linear OOA(467, 2734, F4, 12, 12) (dual of [(2734, 12), 32741, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(467, 16404, F4, 12) (dual of [16404, 16337, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(467, 16405, F4, 12) (dual of [16405, 16338, 13]-code), using
- construction X applied to Ce(12) ⊂ Ce(8) [i] based on
- linear OA(464, 16384, F4, 13) (dual of [16384, 16320, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(443, 16384, F4, 9) (dual of [16384, 16341, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(43, 21, F4, 2) (dual of [21, 18, 3]-code), using
- Hamming code H(3,4) [i]
- construction X applied to Ce(12) ⊂ Ce(8) [i] based on
- discarding factors / shortening the dual code based on linear OA(467, 16405, F4, 12) (dual of [16405, 16338, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(467, 16404, F4, 12) (dual of [16404, 16337, 13]-code), using
(67−12, 67, 14198)-Net over F4 — Digital
Digital (55, 67, 14198)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(467, 14198, F4, 12) (dual of [14198, 14131, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(467, 16405, F4, 12) (dual of [16405, 16338, 13]-code), using
- construction X applied to Ce(12) ⊂ Ce(8) [i] based on
- linear OA(464, 16384, F4, 13) (dual of [16384, 16320, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(443, 16384, F4, 9) (dual of [16384, 16341, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(43, 21, F4, 2) (dual of [21, 18, 3]-code), using
- Hamming code H(3,4) [i]
- construction X applied to Ce(12) ⊂ Ce(8) [i] based on
- discarding factors / shortening the dual code based on linear OA(467, 16405, F4, 12) (dual of [16405, 16338, 13]-code), using
(67−12, 67, 5273557)-Net in Base 4 — Upper bound on s
There is no (55, 67, 5273558)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 21778 081392 049894 443940 846297 910596 848880 > 467 [i]